arxiv:2110.02147v1 [math.ds] 5 oct 2021

DRIFT AND MATRIX COEFFICIENTS FOR DISCRETE GROUP

EXTENSIONS OF COUNTABLE MARKOV SHIFTS

RHIANNON DOUGALL

Abstract. There has been much interest in generalizing Kesten’s criterion

for amenability in terms of a random walk to other contexts, such as deter-mining amenability of a deck covering group by the bottom of the spectrum

of the Laplacian or entropy of the geodesic flow. One outcome of this work

is to generalise the results to so-called discrete group extensions of countableMarkov shifts that satisfy a strong positive recurrence hypothesis. The other

outcome is to further develop the language of unitary representation theory

in this problem, and to bring some of the machinery developed by Coulon–Dougall–Schapira–Tapie [Twisted Patterson-Sullivan measures and applica-

tions to amenability and coverings, arXiv:1809.10881, 2018] to the countable

Markov shift setting. In particular we recast the problem of determining adrop in Gurevic pressure in terms of eventual almost sure decay for matrix

coefficients, and explain that a so-called twisted measure “finds points withthe worst decay.” We are also able locate the results of Dougall–Sharp [Anosov

flows, growth rates on covers and group extensions of subshifts, Inventiones

Mathematicae, 223, 445–483, 2021] within this framework.

1. Introduction

It has been known for some time that one can connect certain structural proper-ties of a group to properties exhibited by a random walk on the group. (A path in arandom walk is the sequence s1 · · · sn ∈ G, for n ∈ N, with each si ∈ G, i ∈ N, inde-pendently sampled according to a probability p on G. Here, we only consider G to becountable and finitely generated.) Kesten’s criterion for the radius of convergenceof a (symmetric) random walk [14] states that if the probability p is symmetric andhas support generating G then the radius of convergence of the (symmetric) randomwalk is equal to one if and only if G is amenable. Motivated by hyperbolic dynam-ics (where independence is not given) it is natural to relax the independence of theproducts s1 · · · sn and instead suppose they are given by a G-extension (also calleda group extension, see subsection 1.1) of a strongly positively recurrent countableMarkov shift (CMS). (A random walk can be understood as a G-extension of a sub-shift of finite type.) It is also natural to remove the symmetry requirement entirely,as we later explain for the example of Anosov flows. The radius of convergence ofthe random walk is naturally realised as the exponential of a Gurevic pressure ofthe G-extension. The notion of pressure is important to geometric examples — itis directly (in cases admitting a symbolic coding) or indirectly (by considering atransliteration in Patterson–Sullivan theory) related to critical exponents of groupsacting on sufficiently negatively curved spaces. There has been much interest inextending the Kesten criterion to dynamical or geometric settings. For example,the early work of Brooks [3] connects amenability of a covering deck group withthe bottom of the spectrum of the Laplacian. In more recent times the advancesbegan in earnest with Stadlbauer’s criterion for the Gurevic pressure of a (weaklysymmetric) group extension [18] which characterises amenability of G in terms ofthe Gurevic pressure of a group extension by G under the following restrictions:the dynamics for the group extension must be transitive, the CMS must satisfy a

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2 RHIANNON DOUGALL

finiteness condition called “the big images and preimages property”, and the Holderpotential and group extension must satisfy a weak symmetry hypothesis.

Amenability of a countable group can be understood in a combinatorial mannerfrom the equivalent notion due to Følner [12]. Yet it can be more enlighteningto phrase in terms of unitary representations. Indeed, to determine whether therecan be a gap in Gurevic pressure over a family of extensions, one is led to con-sider a spectral gap condition for the left regular representation in `2(G/H), forH ≤ G. These were observations of Dougall [7] and Coulon–Dal’bo–Sambusetti [4]who independently gave a gap criterion for extensions of subshifts of finite typewith similar “visibility hypotheses”. (The visibility hypothesis replaces transitivity,which cannot be taken to be uniform in such problems.) We again emphasise thatthe weak symmetry hypothesis is required for one direction of the criterion.

Let us now comment on “symmetry”. A flow is by definition invertible, and inthis way there is a mapping of an increasing-time orbit to a decreasing-time orbit.The geodesic flow has extra symmetry given in the existence of a natural smoothconjugacy of the positive time flow (defined by vector field X ) with the negativetime flow (defined by vector field −X ), preserving both an invariant volume and themeasure of maximal entropy. (This can be encoded to a weak symmetry hypothesisfor a symbolic coding.) Dougall and Sharp [9] study the entropy of an Anosov flowin a covering manifold — in the context of Anosov flows one can already have adrop in entropy for the homology cover [16]. A particular case of their techniquecharacterises the radius of convergence of a finitely supported random walk on anamenable group G as being equal to the radius of convergence for the abelianisationof G, with the restriction that the semigroup G+

p generated by the support ofthe probability p is in fact the whole of G. (In particular there is no symmetryrequirement.) This was proved in the framework of transitive group extensions ofsubshifts of finite type (and in that context G+

p = G implies transitivity of the groupextension). The characterisation of the radius of convergence for an amenable grouphad not been seen before the work of [9]. We expand on the technique of [9] to giveTheorem 3.16, and compare this with a ratio limit theorem (see subsection 2.2).

Finally, we elaborate on the work of Coulon, Dougall, Schapira and Tapie [5],which sought to characterise critical exponents hΓ′ for Γ′ ≤ Γ in terms of thePatterson–Sullivan machinery for the action of Γ on a proper Gromov hyperbolicspace X. They introduced a so-called twisted Patterson–Sullivan measure and char-acterised equality hΓ′ = hΓ in terms of the twisted measure coinciding with theusual Patterson–Sullivan measure for Γ. This geometric setting has local compact-ness that a CMS does not, and so one of our aims is to extend the result to thissetting.

We reinterpret arguments and results of [5] in terms of the existence of a ther-modynamic G-density φ[t] : G→ R, and to show that the parameters at which thisthermodynamic G density φ[t] exists controls the eventual decay of 〈ρ(s1 · · · sn)f, v〉along typical paths s1 · · · sn in a group extension Ts of a CMS. And that diagonalcoefficients associated to φ[t] are naturally related to integrals of functions withrespect a “(twisted) measure” νφc on Σ+. We hope that this exposition will increasethe accessibility of these techniques and grow interest in this novel relation betweenhyperbolic dynamics and unitary representations.

Acknowledgements. This work greatly benefitted from the author’s stay in thetrimester program Dynamics: Topology and Numbers at the Hausdorff ResearchInstitute for Mathematics. The author is grateful to Yves Benoist for the notion ofmatrix coefficients, and to Manfred Einsiedler and Tom Ward for the availability ofa very helpful book draft on unitary representations. The author thanks RichardSharp for many comments in this drafting process. Any errors and misattributions

DRIFT FOR GROUP EXTENSIONS 3

remain the fault of the author. The author also acknowledges the support of theHeilbronn Institute at the University of Bristol.

1.1. Countable Markov shifts and discrete group extensions. More detailfor the definitions is given in Section 12. Let σ : Σ+ → Σ+ be a (one-sided)countable Markov shift with alphabet W1. (In particular Σ+ ⊂ WN

1 but need notbe a full-shift.) As usual, we call a word w ∈ (W1)k admissible if it appears as asubword of some x ∈ Σ+. The two-sided shift is denoted σ : Σ → Σ. Cylindersets given by [w1 · · ·wk] = {x = (x0, x1, . . .) ∈ Σ+ : xi−1 = wi, i = 1, . . . , k} form abasis for the topology on Σ+. We always assume that σ is mixing. Let R : Σ+ →R+ with logR locally Holder continuous (we use multiplicative notation, writingRn(x) = R(x) · · ·R(σn−1x)). We will also view R as a function on Σ. The Gurevicpressure P (logR, σ) is

(1.1) P (logR, σ) = lim supn→∞

1

nlog

∑x∈[B]:σnx=x

Rn(x).

The function R = exp(−P (logR, σ))R has P (log R, σ) = 0, and R satisfies the samerecurrence properties as R. Therefore we always assume that P (logR, σ) = 0.

The countable Markov shift and R : Σ+ → R is said to be positive recurrent(following Sarig [20]) if there is a constant MB with∑

x∈Σ+:σnx=x,x0=B

Rn(x) ∈ [M−1B ,MB ]

for all n ∈ N. We also ask that R is strongly positively recurrent (see subsection12.2 which discusses this condition as it was given by Sarig in [21]), which may beequivalently stated as having that for any B ∈ W1

expP (logR, σ) = 1 > γ(SPR) := lim supn→∞

1

nlog

∑x∈[B]:σnx=x, xi 6=B if i/∈nN

Rn(x).

(For a subshift of finite type, any R with logR being Holder continuous is stronglypositively recurrent — note that we assumed that σ is mixing.)

The shift-invariant probability equilibrium state is denoted µ = µR. (We alsoidentify it as a shift-invariant measure for the two-sided shift space Σ.) We shouldthink of the data of σ,Σ+,W1, R, µ as being fixed and to then vary the skew productsthat follow.

Let G a discrete countable group. A map s : W1 → G defines a s-skew productwith phase space Σ×G and dynamical system

Ts(x, g) = (σx, s(x0)−1g).

The skew product is isomorphic to a countable Markov shift. We also call Ts adiscrete group extension, or G-extension. We lift R to Σ+×G by defining R(x, g) =R(x) (and use the same notation for the function on domain Σ+ and on domainΣ+ ×G). For w ∈ W, w = w1 · · ·wn, define s(w) = s(w1) · · · s(wn). Then

Tns (x, g) = (σx, s(x0 · · ·xn−1)−1g).

The Gurevic pressure P (logR, Ts) is

lim supn→∞

1

nlog

∑(x,e)∈[B]×{e}:Tns (x,e)=(x,e)

Rn(x),

where e is the identity element of G. When Ts is transitive the definition is inde-pendent of B. We always have P (logR, Ts) ≤ P (logR, σ) = 1.

The basic example is given by a random walk p : S → [0, 1] on a group G inwhich we assume S = S−1 generates G. We let Σ+ = SN, R((s0, s1, . . .)) = p(s0)and s(w) = w, once we identify the alphabet with the group elements. Then Ts

4 RHIANNON DOUGALL

describes the p-random walk. The logarithm of the Gurevic pressure coincides withthe radius of convergence of the random walk lim supn→∞ p∗n(e)1/n.

A different example to have in mind is Fa,b, the free group with generators{a, a−1, b, b−1

}, and Σ+ =

{x ∈

{a, a−1, b, b−1

}N: xi+1 6= x−1

i ∀i ∈ N}

, identifying

the alphabet with the group elements. The skew product Σ× Fa,b with s(x0) = x0

walks along geodesics in Fa,b. In this case Ts is not transitive, but it is natural tostudy the quotients.

For s and G fixed we define, for any H EG, sH(x) = s(x)H and correspondingskew product TsH : Σ×G/H → Σ×G/H. Following [7], [4] we say that s satisfiesthe visibility hypothesis if there is a finite set K of G so that every g ∈ G maybe expressed g = k1s(w1) · · · s(wn)k2 for k1, k2 ∈ K and some admissible wordw1 · · ·wn. We expect the visibility hypothesis to be satisfied in settings where thereis a coding of a geometric problem; the visibility hypothesis is used in Coulon–Dal’bo–Sambusetti [4] to characterize a uniform gap in critical exponents of certainisometric actions of word hyperbolic groups, and a more restrictive version is usedin Dougall [7] to characterize a uniform gap in Gurevic entropy for certain geodesicflows. One of the insights of [4] and [7] is that one should understand a family ofskew products in G/H in terms of unitary representation of G.

1.2. Unitary representations and matrix coefficients. A comprehensive texton unitary representations is [1]. Let G be a discrete group. For a (real) Hilbertspace H with inner product 〈·, ·〉H, we denote by U(H) the unitary operatorsfrom H to itself. (Recall that an operator U is unitary if 〈Uv,Uw〉 = 〈v, w〉Hfor all v, w ∈ H.) A unitary representation of G (in H) is a homomorphismρ : G → U(H). The prototypical example is given by the real Hilbert space

`2(G) = `2(G,R) ={f : G→ R :

∑g∈G f(g)2 <∞

}and left regular representa-

tion λ defined by [λ(g)f ](x) = f(g−1x). (One could consider the complex Hilbertspace `2(G,C), but all the representations we consider preserve the real cone andso in this case there is nothing of interest added by complexifying.) For a subgroupH ≤ G we write G/H for the cosets of H in G. We refer to to the unitary represen-tation λH of G in `2(G/H) defined by [λH(g)]f(xH) = f(g−1xH) as the quotientrepresentation. (We are equally tempted to call λH a permutation representationsince one of the key features is that λH permutes an orthonormal basis. See section13.) The scope of this paper is restricted to unitary representations ρ in H that areeither a quotient representation or a countable direct sum of quotient representa-tions. It should be noted that in this paper we rely heavily on the G-invariant coneof non-negative functions `2+(G/H) in `2(G/H).

When G is infinite, there does not exist a vector v ∈ `2(G) invariant by all ofG. A group is said to be amenable if the left regular representation λ has almostinvariant vectors: meaning that for every ε > 0 and every finite set S there is aunit vector v ∈ `2(G) with ‖ρ(s)v − v‖`2(G) ≤ ε for all s ∈ S. One also says thata unitary representation ρ in H weakly contains the trivial representation, written1 � ρ, if for every ε > 0 and every finite set S there is a unit vector v ∈ H with‖ρ(s)v−v‖H ≤ ε for all s ∈ S. If G/Hn, n = 1, 2, . . ., is a sequence of non-amenablequotients (and so 1 ⊀ λHn) one may ask whether 1 � ⊕∞n=1λHn .

We use the notation of convolution throughout the paper: for two functionsf, φ : G→ R we write φ∗f for the function φ∗f(g) =

∫φ(h)f(h−1g)dm, where m is

the counting measure in G (or Haar measure, recalling that G is a countable discretegroup). For a representation π of G write φ ∗π f for the vector

∫φ(h)π(h)fdm.

Frequently one sees π(φ)f or π∗(φ)f ; we prefer φ ∗π f in this setting because theunitary representations are all discrete function spaces `2(G/H). We also use thenotation φ∗ for the function φ∗(g) = φ(g−1); the operator π(φ) =

∑h∈G φ(h)π(h)

has adjoint π(φ∗) (recall that the Hilbert spaces are real).


Fixing the representation ρ of G in H, a vector f ∈ H gives rise to a (diagonal)matrix coefficient ψf : G → R defined by ψf (g) = 〈ρ(g)f, f〉H. Matrix coefficientsplay an important role in the unitary representation theory of groups (where thetheory is at all tractable) such as for compact groups, Abelian groups and semi-simple Lie groups. Here our groups are always countable and equipped with thediscrete topology. When G contains an infinite cyclic subgroup one can find f ∈`2(G) so that ψf is not integrable (ψf /∈ `1(G)) or square integrable (ψf /∈ `2(G)).In a different direction, if φ ∈ `1(G) then

∫ψf (g)φ(g)dm(g) = 〈φ ∗ρ f, f〉`2(G) is

always well-defined and finite, i.e. ψf is integrable with respect to the density φm.

2. Thermodynamic G-densities

Throughout the data of the countable Markov shift and log-Holder function R arefixed, as is s :W1 → G. Recall that we assume P (logR, σ) = 0. The ideology is thatproperties of a unitary representation of G relate to statistics of group extensions,and vice-versa. To this end, we need a mathematical object by which to revealthis relationship. We introduction a “thermodynamic G-density”, a family of non-negative functions φ[t] : G → G associated to a convergence parameter t ∈ R, andshow a relationship to certain statistics of a unitary representation.

Remark 2.1. The emphasis is to begin with the function φ[t] and then for eachunitary representation ρ in H and f ∈ H consider the vectors φ[t] ∗ρ f , rather thango directly to the operator ρ∗(φ[t]). The operator ρ∗(φ[t]) appears (in a differentguise) in two other contexts. In the work of [5] the twisted Poincare series could betransliterated to ρ∗(φ[t]). When the underlying skew product describes a randomwalk we have that

∑a,b∈W1

ρ∗(φa,b[t]) coincides with the Neumann series of the

random walk operator (for t large enough). One might consider the Neumannseries for the transfer operator for Ts but this has disadvantage that one has toworry about the whole space Σ+ ×G — this is the topic of future work.

Recall that we write WA,an for admissible words w = w1 · · ·wn of length n with

first letter w1 = A and last letter wn = a.

Definition 2.2 (Thermodynamic G-density). Let A, a ∈ W1 and x ∈ σ[a]. Let c :

N→ [0,∞) with subexponential growth. Define the partial sums φA,ac;≤N [t] : G→ Gby

φA,ac;≤N [t](g) =

N∑n=1

t−ncn∑

w∈WA,an :s(w)=g

Rn(wx).

For any t > 0 such that supN φA,ac;≤N [t](g) < ∞ for each g ∈ G, we call φA,ac [t] :

G→ G

φA,ac [t](g) =

∞∑n=1

t−n∑

w∈WA,an :s(w)=g

Rn(wx).

a thermodynamic G-density.

In the case that c is identically 1 we abbreviate to φA,a≤N [t](g) and φA,a[t] respec-

tively. If Σ+ is not compact we insist that aA is admissible, and x ∈ [A].

We continue the exposition for a fixed A, a, x, and use the shorter notationφ[t] = φA,a[t] (and φc[t] = φA,ac [t]). When Ts is transitive (and logR is Holder)the convergence parameters that follow are independent of these choices. (Or ifρ corresponds to the quotient representation `2(G/H) then we ask for TsH to betransitive.) In the case that H = `2(G/H) we write δh to denote the indicatorfunction at a coset h ∈ G/H, and δe denotes the indicator function at the cosetcorresponding to H.

6 RHIANNON DOUGALL

Let ρ and H be a countable sum of quotient representations of G. (We en-compass the group extension sH : W1 → G/H using the quotient representationfor H E G). We make use of the cone of non-negative functions `2+(G/H) ={f ∈ `2(G/H) : ∀g ∈ Gf(g) ≥ 0

}; and denote H+ the corresponding cone in H.

Recall that a vector f ∈ H gives rise to a matrix coefficient h 7→ 〈ρ(h)f, f〉H. Wedefine convergence parameters according to the integrability of matrix coefficientsin the following way.

Definition 2.3 (Hierarchy of convergence parameters). For f ∈ H+ denote

γ(f) = inf

{t > 0 : sup

N∈N

∫〈ρ(h)f, f〉Hdφ≤N [t](h) <∞

},

γ(φ∗ ∗ρ f) = inf

{t > 0 : sup

N∈N

∫〈ρ(h)f, f〉Hdφ≤N [t] ∗ φ≤N [t](h) <∞

}= inf

{t > 0 : sup

N∈N〈φ≤N [t] ∗ρ f, φ∗≤N [t] ∗ρ f〉H <∞

},

γ(φ ∗ρ f) = inf

{t > 0 : sup

N∈N

∫〈ρ(h)f, f〉Hdφ∗≤N [t] ∗ φ≤N [t](h) <∞

}= inf

{t > 0 : sup

N∈N〈φ≤N [t] ∗ρ f, φ≤N [t] ∗ρ f〉H <∞

}.

We say that the pressure of f is − log γ(f). We say that γ(φ ∗ρ f) is the decayexponent for f .

We justify the terminology by Proposition 3.1 and Lemma 3.5. Let us commenton the hierarchy of convergence — the verification of the statements follows later.We simplify here to H = `2(G), and so f ∈ `2+(G).

(1) The hierarchy is increasing: γ(f) ≤ γ(φ∗ ∗ f) ≤ γ(φ ∗ f).(2) For t > γ(SPR), φc;≤N [t] belongs to `1(G).(3) For t > γ(δe), φc[t] = limN→∞ φc;≤N [t] is well-defined as an element of GR.(4) For t > γ(φ ∗ f), φc[t] ∗ f ∈ `2(G).

Most textbooks featuring convolutions of functions explain that for f ∈ `2(G), wehave that the convolution v ∗ f satisfies v ∗ f ∈ `2(G) if v ∈ `1(G). The conditionthat v ∈ `1(G) is not optimal — later Theorem 3.13 will tell us that if G is non-amenable there is t < 1 so that φ[t] = φ[t] ∗ δe ∈ `2(G) despite φ[t] /∈ `1(G). We usethe positivity of the functions throughout this paper. We check the next lemma inSection 13.

Lemma 2.4. Let f ∈ H+. If supN∈N ‖φc;≤N [t]∗ρf‖ <∞ then φc;≤N [t]∗ρf stronglyconverges to φc[t] ∗ρ f as N →∞, and φc[t] ∗ρ f ∈ H+.

Remark 2.5. (1) The vector norm ‖φ[t]‖`2(G) should not be confused with the

operator norm ‖ρ∗(φ[t])‖op := supf 6=0

‖φ[t]∗f‖`2(G)

‖f‖`2(G).

(2) We only make use of the unitarity of the representation ρ for results involv-ing the largest parameter γ(φ∗ρf). A hidden consequence of unitarity is thestrong convergence of φ≤N [t]; in particular 〈φ≤N [t], φ≤N [t]〉 → 〈φ[t], φ[t]〉as N → ∞. For t > γ(φ∗ ∗ρ f) we know only that 〈φ≤N [t], φ∗≤N [t]〉 has alimit, and our manipulations only utilise the homomorphism property of ρand that ρ preserves the non-negative cone.

Secondly, we explain the effect of the choice of A, a. For two vectors v, w ∈ `2+(G)we say that v ≤ w if v(g) ≤ w(g) for every g ∈ G.


Lemma 2.6. For each B there are a constants Const.(B), KB and a group elementshB,gB with

(2.1) Const.(B)−1δhB ∗ φA,ac;≤N−KB [t] ≤ φB,ac;≤N [t] ≤ Const.(B)δgB ∗ φ

A,ac;≤N+KB

[t].

If Ts is transitive then we may assume gB = e = hB.

In Section 13 we also check other basic facts, such as the connectedness of theparameters t for which the defining integrals converge, and that transitivity implies0 < γ(f) = γ(φ∗ ∗ρ f).

2.1. Examples. For the Abelian group Z and vector f = δ0 we have∫〈ρ(k)δ0, δ0〉`2(Z)dφ≤N [t](k) =

∫ N∑n=1

t−n∑

w∈WA,an

Rn(wx) exp(2πis(w)α)dα

=:

∫ζ≤N (t, α)dα

The implied series ζ(t, α) converges for all t ≥ | exp(P (logR + 2πiαs, σ))|. Att = exp(P (logR, σ)) = 1 the series ζ(1, 0) = ∞, but this does not guarantee thatthe integral over all α is infinite. For a transitive random walk on Z that does nothave zero mean we have γ(δ0) < 1 — see [26].

Let S be a generating set for a group G, and p : S → [0, 1]. Then for theG-extension corresponding to the random walk we have

φ[t](g) =∑n∈N

t−np∗n(g).

In the case that G = Fa,b, S ={a, b, a−1, b−1

}and p = 1/4, one can compute

asymptotics for p∗2n(g) (see [26]) and in particular we deduce that γ(δe) =√

3/2.This example is elaborated on in section 4.

2.2. Related notions: ratio limit theorems and local limit theorems. It isworth mentioning related concepts in the context of random walks since we findour constructions to be reminiscent. We quote a ratio limit theorem from the bookof Woess [26]. Set γ = limn→∞(p∗n(e))1/n and note that this coincides with ourdefinition of γ(δe).

Proposition 2.7 (Ratio Limit Theorem, Theorem 5.6 (and Corollary 5.8) of [26]).Suppose that G is Abelian (or nilpotent) and p is a finitely supported measure whosesupport generates G as a semigroup. Then there is a unique γ-harmonic functionΨ : G→ R with

limn→∞

p∗n(x)

p∗n(e)=

1

Ψ(x)

for all x ∈ G.

Let us elaborate on the case where G is nilpotent: the assumption that thesupport of p generates G as a semigroup forces that the harmonic function Ψ forG is also harmonic for the abelianisation of G. Recent work of Benoist [2] finds anew harmonic function for the Heisenberg group for a probability whose supportdoes not generate G as a semigroup (and the ratio limit theorem stated in [2] is ofa different nature than what is stated in Proposition 2.7). In section 4 we discussfurther the setting of a random walk when p is finitely supported, (aperiodic,) andwith the assumption that the semigroup generated by the support is the whole ofG (this last assumption implies transitivity of the group extension). When G isAbelian we show in Proposition 4.2 how to directly recover the function Ψ (but notthe ratio limit theorem) from Lemma 3.9. Theorem 3.16 also recovers the functionΨ and, notably, is valid even for amenable groups.

8 RHIANNON DOUGALL

A stronger result to mention is a local limit theorem (to keep this discussion lightwe do not mention the local limit theorem for Abelian groups). For the free groupFr on r generators one can find CΨ with

lim supn→∞

p∗n(x)

γnn−3/2= CΨ(x) > 0

for all x ∈ Fr (Theorem 6.8 of [26] and attributed to Lalley, and also in this isotropiccase to Picardello [23]). We give a calculation to explain how one recovers Ψ (butnot the local limit theorem) in the case of the simple random walk in Section 4.

3. Results

3.1. Preliminaries. We motivate the study of the thermodynamic G-density byits connection to growth statistics for discrete group extensions.

Let us begin with Gurevic pressure and notion of decay of matrix coefficientsalong µ typical paths.

Proposition 3.1. Assume that Ts is transitive. If expP (logR, Ts) > γ(SPR) then

γ(δe) = expP (logR, Ts),

where δe is the indicator function at the identity for the left regular representationλ in `2(G).

The proof is found in Section 12.A random walk on a group is said to be transient (see [26]) if the probability

p : G → [0, 1] has∑∞n=1 p

∗n(e) < ∞. We reimagine this as a statement aboutdecay of pairs of vectors in `2(G). Ultimately, the idea is that if the left regularrepresentation λ in `2(G) does not weakly contain the trivial representation (Gis non-amenable) then pairs of vectors are forced to have a so-called statistical-dynamical eventual decay.

Definition 3.2. Assume Σ is compact. We say that vectors f, v ∈ H+ have a(statistical-dynamical) eventual decay if there is γ < 1 so that for µ almost everyx ∈ Σ there is N with

n ≥ N =⇒ 〈ρ(s(x−n · · ·x−1))f, v〉H ≤ γn.

We say that ρ,H has uniform (statistical-dynamical) eventual decay exponent ifwe can take γ < γ0 for some γ0 < 1 independent of f, v ∈ H+.

Remark 3.3. One may also consider decay for the products ρ(s(x1 · · ·xn))f (andindeed we do in Lemma 3.11 as the measure there is only defined on the one-sidedshift). Lemma 3.5 is also valid for products ρ(s(x1 · · ·xn))f .

Remark 3.4. Note that the Definition 3.2 is scalar independent. The significanceof the decay statement is given by the arbitrariness of v. We illustrate this with theexample of a symmetric random walk on a group.

Consider f = δe and v = δe. The property that f = δe and v = δe have almostsure eventual γ-decay is equivalent to asking that the µ measure of points (x, e)that Ts-return infinitely often to Σ+×G is zero. The (assumed symmetric) randomwalk on Zd is transient for d > 2 [26], and so f = δ0, v = δ0 have eventual decayfor any γ < 1.

On the other hand, in Zd and f = δ0 we can find a function v for which f ,v donot have statistical-dynamical decay (with respect to a symmetric random walk).

The function vt(k) = |k|−t belongs to `2(Z) if t > 12 ; and similarly vt(~k) = ‖~k‖−t1

belongs to `2(Zd) if d− 2t < 2. Suppose that s takes values in the unit ‖ · ‖1-ball.

Let γ < 1. Choose M = Mγ so that vt(~h) > γ‖~h‖1 for ~h in the complement to the

‖ · ‖1-ball of radius M . Then 〈λ(s(x−n · · ·x−1))f, vt〉`2(Zd) > γn for any n > M .


In an arbitrary amenable group, with some extra work we can show that for everyγ there are vectors f = vγ and v = vγ which have cannot have decay rate fasterthan γ. In this way, for G amenable, λ, `2(G) does not have uniform (statistical-dynamical) eventual decay. (The argument presented for Zd relied on polynomialgrowth and gave the stronger conclusion of exhibiting vectors not having eventualdecay.) See Lemma 4.1.

Lemma 3.5. Assume Σ is compact. Let f ∈ H+. If γ(φ ∗ρ f) < 1 then for everyγ ∈ (γ(φ ∗ρ f), 1) and every v ∈ H+ we have that for µ almost every x ∈ Σ there isN with

n ≥ N =⇒ 〈ρ(s(x−n · · ·x−1))f, v〉H ≤ γn

Remark 3.6. In the case that Σ is not compact one should expect a restriction onx returning to a small open set.

Proof of Lemma 3.5. Let f, v ∈ H+ be arbitrary. We denote the sets

En = {x ∈ Σ : 〈s(x−n) · · · s(x−1)f, v〉H ≥ γn} ,

lim supEn = {x ∈ Σ : 〈s(x−n) · · · s(x−1)f, v〉H ≥ γn, n i.o.} .Our goal is to use the Borel-Cantelli Lemma to deduce that lim supEn has zero µmeasure. Negative-coordinate cylinder sets are denoted

[un · · ·u−1.] = {x ∈ Σ : x−i = u−i, i = 1, · · ·n} .

The Gibbs property 12.13 states that there is a constant C > 0 with

µ([w.]) ≤ CRn(wy)

for any w of length n and y ∈ σn[w.]. We compute

µ(En) =∑

A,a∈W1

∑w∈WA,a

n :〈s(w1)···s(wn)v,f〉H≥γn

µ([w1 · · ·wn.])

≤∑

A,a∈W1

∑w∈WA,a

n :〈s(w1)···s(wn)f,v〉H≥γn

µ([w1 · · ·wn.])γ−n〈ρ(s(w1) · · · s(wn))f, v〉H

≤ C∑

A,a∈W1

∑wWA,a

n

Rn(w1 · · ·wnx)γ−n〈ρ(s(w1) · · · s(wn))f, v〉H

≤ C∑

A,a∈W1

〈φA,an [γ] ∗ρ f, v〉H,

with φA,an [t](g) :=∑wWA,a

n :s(w)=g t−nRn(w1 · · ·wnx). Then

∞∑n=1

µ(En) ≤∑

A,a∈W1

〈φA,a[γ] ∗ρ f, v〉H ≤∑

A,a∈W1

‖φA,a[γ] ∗ρ f‖H‖v‖H.

By Lemma 2.4 we know that ‖φA,a[γ] ∗ρ f‖H is finite for γ > γ(φ ∗ρ f), thus givingthe convergence case of the Borel-Cantelli Lemma as desired. �

3.2. (Limits of) matrix coefficients of the thermodynamic G-density andtwisted measures. We refer the reader to Subsection 7.1 and Definition 7.1 for thedefinition of a measure twisted by a generalised multiplicative cocycle. We are notable to work with φ[t] but rather must increase it with a slowly increasing function.

Lemma 3.7. Let f ∈ H+. There is a slowly increasing function c : N → ∞ andsequence tk → γ(φ ∗ρ f) as k →∞ with the following. The function Υc : G→ R

Υc(g) = limk→∞

〈ρ(g)φc[tk] ∗ρ f, φc[tk] ∗ρ f〉H〈φc[tk] ∗ρ f, φc[tk] ∗ρ f〉H

.

10 RHIANNON DOUGALL

is well-defined, and there is a twisted measure νφ∗ρfc on Σ+ (that is finite on cylin-

ders) with

(γ(φ ∗ρ f))nνφ∗ρfc (R−1n 1[w]) = Υc(s(w))

for all w ∈ Wa,An+1.

Remark 3.8. The measure in 3.7 can be thought of as being analogous to thetwisted measure in [5]. However, there are some caveats. The setting of [5] isgeometric SPR, and they define the measure according to a sequence fk maximisingthe twisted operator, whereas in this exposition fk = f . In [5] the one-sided measureis turned into a group action invariant current by a generalised product with itself.

In the absence of symmetry νφ∗ρfc is not easily related to a shift invariant measure.

(For an example with symmetry see also equation 4.2.)

Recall that logR is assumed to be (locally) Lipschitz with respect to the θ-metric(see equation 12.1).

Lemma 3.9. Assume Σ+ is compact, and assume Ts is transitive. Let f ∈ H+.There is a slowly increasing function c : N → ∞ and sequence tk → γ as k → ∞with the following. The function Υc,∗ : G→ R

Υc,∗(g) = lim supk→∞

limN→∞

〈ρ(g)φc;≤N [tk] ∗ρ f, φ∗c;≤N [tk] ∗ρ f〉H〈φc;≤N [tk] ∗ρ f, φ∗c;≤N [tk] ∗ρ f〉H

is well-defined, and there is a shift invariant finite measure µφ∗∗ρfc on Σ+

(3.1) C−θw∧x

Υc,∗(s(w)) ≤ (γ(f))nµφ∗∗ρfc (R−1

n 1[w]) ≤ Cθw∧x

Υc,∗(s(w))

for some constant C > 1.

Remark 3.10. (1) The c appearing in Lemmas 3.7 and 3.9 can be different.(And indeed the γ can be different.)

(2) The functions Υc,Υc,∗ implicitly depend on the letters A, a for which φ =φA,a.

(3) The function Υc is defined by a limit (along tk) of matrix coefficients. Wecan check that Υc is positive definite (

∑i,j∈I αiαjΥc(g

−1j gi) ≥ 0), from

which Υc is a matrix coefficient for a unitary representation (see [23] or [10]).Section 14 gives an explicit example.

We give an interpretation that the νφ∗ρfc measure finds points with the slowest

decay exponent — for a reversed ordering of the products in Lemma 3.5. We makethis more precise in the case of a symmetric random walk on the free group wherewe are able to express section Υc(g) in terms of an exponential — see Section 4.In general we can only say that a typical point has decay bounded by Υc(g). Andthat Υc(g) is typically decays no faster than γ.

Lemma 3.11. Assume Σ+ is compact and that Ts is transitive. Let f ∈ H+. If

γ(φ ∗ρ f) < 1 then for every ε > 0 and every v ∈ H+ we have that for νφ∗ρfc almost

every x ∈ Σ+ there is N with

n ≥ N =⇒ 〈ρ(s(x1 · · ·xn))f, v〉H ≤ (1− ε)−nΥc(s(x1 · · ·xn))

In addition for νφ∗ρfc almost every x ∈ Σ+ there is N with

(3.2) n ≥ N =⇒ Υc(s(x1 · · ·xn)) ≥ n−2(γ(φ ∗ρ f))n.

Remark 3.12. (1) We present a calculation for a random walk on the freegroup (see Section 4). In this case γn = 2n3−n/2 whereas it can be shownthat Υc has infinitely many g with Υc(g) ≥ 3−|g|/2. In fact a certain linearcombination relating to Υc(g) (taking into account different choices of A, a)is equal to (1 + |g|/2)3−|g|/2. Then 3.3 implies that a νφc typical point x has


that s(x1 · · ·xn) is not reduced in a quantitative way, which we comparewith the known drift for µ random walk.

(2) If γ(φ ∗ρ f) = γ(f) then, upon identifying µφ∗∗ρfc with a shift invariant

measure on Σ we can show that for every γ(f) < γ and every v ∈ H+ we

have that for µφ∗∗ρfc almost every x ∈ Σ there is N with

n ≥ N =⇒ 〈ρ(s(x−n · · ·x−1))f, v〉H ≤ (1− ε)−nΥc,∗(s(x−n · · ·x−1)).

and

n ≥ N =⇒ 〈ρ(s(x−n · · ·x−1))f, v〉H ≤ (1− ε)−nΥc,∗(s(x1 · · ·xn)).

In addition for νφ∗ρfc almost every x ∈ Σ+ there is N with

(3.3) n ≥ N =⇒ Υc(s(x−n · · ·x−1)) ≥ n−2(γ(φ ∗ρ f))n.

Proof of Lemma 3.11. Let f, v ∈ H+ be arbitrary. For brevity write γ = γ(φ ∗ρ f).Set

En ={x ∈ Σ : 〈ρ(s(x1 · · ·xn))f, v〉H ≥ Υc(s(x1 · · ·xn))(1− ε)−n

},

lim supEn ={x : 〈ρ(s(x1 · · ·xn))f, v〉H ≥ Υc(s(x1 · · ·xn))(1− ε)−n, n i.o.

}.

Lemma 3.7 is stated only for cylinders wA, however Lemma 7.11 upgrades to

νφ∗ρfc ([u]) ≤ CRn(uy)Υc(s(u))γ−n

for any u of length n and y ∈ σn[u]. We compute

νφ∗ρfc (En) =∑

A,a∈W1

∑w∈WA,a

n :

〈ρ(s(w1···wn))v,f〉H≥Υc(s(w))(1−ε)−n

νφ∗ρfc ([w1 · · ·wn])

≤∑

A,a∈W1

∑w∈WA,a

n :

〈ρ(s(w))f,v〉H≥Υc(s(w))(1−ε)−n

νφ∗ρfc ([w])

Υc(s(w))(1− ε)n〈ρ(s(w))f, v〉H

≤ C∑

A,a∈W1

∑wWA,a

n

γ−n(1− ε)nRn(wx)〈ρ(s(w))f, v〉H

≤ C∑

A,a∈W1

〈φA,an [γ(1− ε)−1]f, v〉H.

Since γ(1 − ε)−1 > γ(φ ∗ρ f) the convergence case of the Borel-Cantelli Lemmafollows.

For the second part set

En ={x ∈ Σ+ : Υc(s(x1 · · ·xn)) ≤ γnn−2

}.

We have

νφ∗ρfc (En) =∑

A,a∈W1

∑w∈WA,a

n :

Υc(s(w))≤n−2γn

νφ∗ρfc ([w])

≤∑

A,a∈W1

∑w∈WA,a

n :

Υc(s(w))≤n−2γn

CRn(wy)Υc(s(w))γ−n

≤ C∑

A,a∈W1

∑wWA,a

n

Rn(wx)1

n2.

The series∑∞n=1 n

−2 is summable and so we conclude the convergence case of theBorel-Cantelli Lemma. �

12 RHIANNON DOUGALL

3.3. Theorems. Lemma 3.7 is the main ingredient to our first two theorems.

Theorem 3.13. Assume that Ts is transitive. We have the following:

• If G is non-amenable then P (logR, Ts) < 0.• If ρ does not weakly contain the trivial representation then ρ, H has uniform

eventual decay exponent γ0 = sup0 6=f∈H+γ(φ ∗ρ f) < 1.

Theorem 3.14. Assume that s satisfies the visibility hypothesis. Let F be a familyof normal subgroups of G. We have

1 ⊀⊕H∈F

`2(G/H) =⇒ sup {P (logR, TsH ) : H ∈ F} < 0.

Remark 3.15. (1) In section 4 we give an example to see that the decay state-ment of Theorem 3.13 is non-trivial.

(2) Theorem 3.14 only uses normality for the definition of TsH as a group exten-sion. Transitivity is not required as the proof relies on the study of γ(φ ∗ f)for f the countable sum of `2(G/H).

Assume now that Σ is compact. For amenable groups the work of [9] showsthat the Gurevic pressure P (logR, Ts) is equal P (logR+ ψ) for a unique real one-dimensional representation π : G → R and ψ = π ◦ s. (In other words: let ψab bethe composition of s with G→ G/[G : G] ∼= Zd⊕F → Zd, with F the finite torsiongroup. Then there is a unique ξ ∈ R that determines ψ(x) = 〈ξ, ψab(x)〉Rd .)

Theorem 3.16. Assume Σ+ is compact and assume Ts is transitive. If G isamenable then the measure µφ

∗∗δec from Lemma 3.9 is the equilibrium state for

R expψ. In particular

Υc,∗(g) = expψ(g).

Remark 3.17. In section 4 we discuss further the setting of a random walk whenp is finitely supported, (aperiodic,) and with the assumption that the semigroupgenerated by the support is the whole of G (this last assumption implies transitivityof the group extension). When G is Abelian we show in Proposition 4.2 how torecover the function Ψ of the ratio limit theorem as stated in Proposition 2.7 (butnot recover the ratio limit theorem) from Lemma 3.9 by

Υc,∗(g) =1

Ψ(g).

Theorem 3.16, notably, is valid even for amenable groups.

4. (Symmetric) random walks

We vary G, a (symmetric) finite generating set S, and the probability p : S →[0, 1], but fix that the structure of the group extension to describes the randomwalk. Namely, Σ = SZ is the full shift on a (symmetric) finite generating set Sof G, R = p : S → [0, 1], and s idenifies the formal letter a ∈ W1 with the groupelement it represents. We assume that the semigroup generated by S is equal to G.In this way the group extension is always transitive.

The structure of a random walk is useful as

(4.1)∑

B,b∈W1

〈φB,b[t], δe〉 =

∞∑n=1

p∗n

(e)t−n.

When S is symmetric φ and φ∗ coincide. Yet νφc need not coincide with µφ∗

c as

νφc ([b]) = limq→∞

〈φb,ac [tq], φA,ac [tq]〉

〈φA,ac [tq], φA,ac [tq]〉


whereas

µφc ([b]) = limq→∞

〈φb,ac [tq],∑D∈W1

φD,ac [tq]〉〈φA,ac [tq], φ

A,ac [tq]〉

.

Let νA,a;D,ac be the measure with

νA,a;D,ac ([b]) = lim

q→∞

〈φb,ac [tq], φD,ac [tq]〉

〈φA,ac [tq], φA,ac [tq]〉

.

Let µdc be the measure with

µdc([b]) = limq→∞

〈φb,dc [tq],∑B∈W1

φB,ac [tq]〉〈φA,ac [tq], φ

A,ac [tq]〉

.

Then

(4.2)∑D∈W1

νa;D,ac = µac ,

with mass

CAa,a =∑D∈W1

νa;D,ac (1) = lim

q→∞

〈∑b∈W1

φb,ac [tq],∑D∈W1

φD,ac [tq]〉〈φA,ac [tq], φ

A,ac [tq]〉

.

Let us also point out that ΥB,b;D,dc (g) = limq→∞

〈ρ(g)φB,bc [t],φD,dc [t]〉〈φA,ac [t],φA,ac [t]〉

has

(4.3) ΥB,b;D,dc (s(w)) = γ|w|p−|w|νb;D,dc ([wB]).

(See also equation 4.5.)The condition 1 � ρ can be characterized by operator norms: namely 1 � ρ

implies that for any F : G → R we have ‖ρ∗(F )‖op = ‖1∗(F )‖op = ‖F‖`1(G). Inparticular ‖ρ∗(φ[t])‖op → ∞ as t → 1, but this is weaker than the existence of fwith γ(φ ∗ f) = 1.

We first check the non-triviality of the decay statement by giving an amenableexample that does not have decay.

Lemma 4.1. Assume that S is symmetric. Suppose that G is amenable and ρ = λthe left regular representation. Then for every γ < 1 there is v = vγ so that

〈ρ(s(w))v, v〉 ≥ γ|w| for w with |w| sufficiently large.

Proof. As G is amenable and p symmetric we deduce from equation 4.1 that γ(φ ∗δe) = 1. Then (using for instance equation 4.2) µφc coincides with µ = µp theequilibrium state for p. Using Lemma 3.7 we have

µφc ([aWA]) = p|W | limq→∞

〈ρ(s(W )φc[tq], φc[tq]〉〈φc[tq], φc[tq]〉

On the other hand, since µ([aWA]) = p|W |+2 we must have

p2 = limq→∞

〈ρ(s(W ))φc[tq], φc[tq]〉〈φc[tq], φc[tq]〉

.

Let η < 1 which will be determined in terms of γ < 1. Define

vη =

∞∑q=1

φc[tq]

‖φc[tq]‖2

14 RHIANNON DOUGALL

For any g = s(W ) we have

〈ρ(g)vη, vη〉`2(G) =∑n∈N

∑m∈N

ηnηm〈ρ(g)φc[tn], φc[tm]〉`2(G)

‖φc[tn]‖`2(G)‖φc[tm]‖`2(G)

≥∑n∈N

(η2)n〈ρ(g)φc[tn], φc[tn]〉`2(G)

‖φc[tn]‖2`2(G)

≥∑

n∈N:n≥|w|

(η2)np2

We assume η is sufficiently close to 1 so that∑n∈N:n≥k(η2)np2 > γk for all k larger

than some K. �

We now explain directly from the definitions how to obtain the harmonic functionfrom the Ratio Limit Theorem.

Proposition 4.2. If G is Abelian then

Υc,∗(g) =1

Ψ(g)

for Ψ in the Ratio Limit Theorem 2.7.

Proof. When G is Abelian we can rearrange terms

Υc,∗(g) = lim supk→∞

limN→∞

〈φc;≤N [tk] ∗ φc;≤N [tk] ∗ρ f, ρ(g)−1f〉H〈φc;≤N [tk] ∗ φc;≤N [tk] ∗ρ f, f〉H

In particular given that we assume H = `2(G), f = δe and that the group extensionis for a random walk, we have

(4.4) Υc,∗(g) = lim supk→∞

limN→∞

〈φc∗c;≤N [tk], δg〉H〈φc∗c;≤N [tk], δe〉H

.

(In checking this we use Equation 6.1 and divergence of the denominator.)Use the notation F1 �C F2 to mean C−1F1 ≤ F2 ≤ CF1. The ratio limit theorem

for random walks (Proposition 2.7) says that there are εn(g) → 0 as n → ∞ withΨ(g)p∗n(g) = (1 + εn(g))p∗n(e). Substituting this into equation 4.4 gives

Υc,∗(g) �(1+ε′M (g))1

Ψ(g)lim supk→∞

∑n>M (c ∗ c)nt−nk p∗n(e)∑n>M (c ∗ c)nt−nk p∗n(e)

,

with ε′M (g) = supn>M |εn(g)|, and again using divergence of the denominator and

boundedness of∑n≤M (c ∗ c)nt−nk p∗n(e) for M fixed. Since M is arbitrary and

ε′M (g)→ 0 the conclusion follows. �

We now specialise to the free group Fa,b with free basis S ={a, a−1, b, b−1

}.

Consider the uniform random walk p(s) = 4−1 for each s ∈ S. One can determinethe radius of convergence in this case. (To do the calculation, it is helpful torepresent returns to the identity in terms of Catalan numbers.) See also [23].

Proposition 4.3 (Theorem 3 in [15]; Corollary 3.6. in [26]). For SN, s, p given bythe simple random walk on the free group G = Fa,b we have

γ(δe) =

√3

2.

There is a special class of functions Ψ : Fa,b → R called the spherical functions— the reference we use for this discussion is the exposition of Figa-Talamancaand Picardello [11]. By definition a spherical function Ψ : Fa,b → R is constanton every sphere (is radial) and is multiplicative on the convolutional algebra ofradial functions (and one tends to normalise so that Ψ(e) = 1). Alternatively, thespherical functions are precisely eigenfunctions p ∗ Ψ = γ(Ψ)Ψ with p the uniform


probability on the free basis, and can be characterized in terms the Poisson-Kernel(we explain a particular case in section 14). Our construction of φc[t] = φA,ac [t] andΥc = ΥA,a;A,a

c is not optimized to be radial, but we claim that there is a naturallinear combination that is spherical for the free group. As we have specialised tothe symmetric setting we have Υc = Υc,∗.

Lemma 4.4. For any c the function Υ =∑B,b,D,d∈W1

ΥB,b;D,dc is spherical, and

moreover

Υ(g) = C

(1 +|g|2

)3−|g|/2

for some constant C > 0.

Proof. It is easy to see that Υ is constant on spheres from

# {wn ∈ Sn, um ∈ Sm : gwn = um} = # {wn ∈ Sn, um ∈ Sm : gwn = um} ,for each g, g with |g| = |g|.

If a function Ψ, constant spheres, satisfies the equation p ∗ Ψ = rΨ then Ψ(g)

is completely determined by an induction from the identity term Ψ(e). We already

know from [11] that the function Ψ(g) =(

1 + |g|2

)3−|g|/2 has p ∗Ψ = 2−131/2Ψ. It

is enough to check that Υ has p ∗Υ = 2−131/2Υ.Equation 4.3 implies that

(4.5) Υ(s(w)) =∑

B,a′,D,a′′∈W1

γ|w|p−|w|νa′;D,a′′([wB]).

Consequently for any s ∈ S and letting g = s(w) we have

Υ(sg) = γp−1∑

B,a′,D,a′′∈W1

γ|w|p−|w|νa′;D,a′′([swB]).

But also ∑D∈W1

νa′;D,a′′([swB]) = νa

′;s−1,a′′([wB]).

In this way∑s∈S

p(s)Υ(sg) = γ∑

B,a′,a′′∈W1

∑s∈S

γ|w|p−|w|νa′;s−1,a′′([wB]) = γΥ(g).

It follows that Υ is a constant multiple of Ψ. �

Remark 4.5. (1) We see in the proof of Lemma 4.4 that the coincidence be-tween Υ and a constant multiple of Ψ is entirely due to the eigenvalue.There is not an immediate comparison with the local limit theorem since itconcerns 〈ρ(g)φ[t] ∗ δe, δe〉 and not 〈ρ(g)φ[t] ∗ δe, φ[t] ∗ δe〉.

(2) For the unacquainted, we include in section 14 that the curious-looking

function Ψ(g) =(

1 + |g|2

)3−|g|/2 is a matrix coefficient for the boundary

representation.

Lemma 4.4 has consequences for the lower and upper bound on rate of decay.

Corollary 4.6. For each ε > 0, a µφc typical point has

〈ρ(s(x−n · · ·x−1))f, v〉 ≤ |s(x−n · · ·x−1)|√

3|s(x−n···x−1)|

(1 + ε)n√

3n

2n

for n sufficiently large (depending on x).For each ε > 0, a µ typical point has

〈ρ(s(x−n · · ·x−1))f, v〉 ≤ (1 + ε)n√

3n

2n

for n sufficiently large (depending on x).

16 RHIANNON DOUGALL

It is well-known that the drift l(p) for the simple random walk p on the freegroup on 2 generators is equal to 1/2 (using that l(p) =

∫b(g, ξ)dp(g)dν(ξ) with b

the Busemann additive cocycle — see Proposition 2.2. of [13].). That says that fora µ typical x we have

limN→∞

1

n|s(x−n · · ·x−1)| = 1

2.

This gives in particular a lower bound on the amount of cancellation in s(x−n · · ·x−1).For a µφc typical point we have quantitatively more cancellation along a subsequence.

Lemma 4.7. A µφc typical point x has

lim supn→∞

1

n|s(x−n · · ·x−1)| ≤ 2 log 2− log 3

log 3<

1

2.

Proof. We know from Lemma 3.11 together with Remark 3.12, and the symmetryof the probability, that a µφ

∗

c typical x has

Υc(s(x−n · · ·x−1)) ≥√

3n2−nn−2

for all n > Nx.Since Υ is a linear combination of ΥB,b;D,d

c,∗ it follows in particular for ΥA,a;A,ac,∗

that there is a constant C > 0 for which a µφ∗

c typical x has

Υ(s(x−n · · ·x−1)) ≥ CΥA.a;A,ac (s(x−n · · ·x−1)) ≥

√3n2−nn−2

for n larger than some Nx.Now using the identity for Υ and writing g = s(x−n · · ·x−1), this gives for every

ε > 0 and N sufficiently large

(√

3)−(n+|g|) ≥ (1 + ε)n2−n

Equivalently

|g|2

log 3 +n

2log 3 ≤ n log 2 + n log(1 + ε).

Let εk be a sequence with εk → 0 as k →∞. Then we have a full measure set Efor which

lim supn→∞

1

n|s(x−n · · ·x−1)| ≤ 2 log 2− log 3

log 3+

log(1 + εk)

log 3.

for every k. The conclusion follows. (One also checks by computation that 2 log 2−log 3log 3 <

12 .) �

5. One sided twisted measures: preliminary constructions

In sections 2 and 3 the notation is descriptive of the objects. We consider afixed f ∈ H+ and a fixed slowly increasing function c (that has to satisfy certainproperties), and so their dependency is suppressed in the notation that follows. Wede-clutter notation by writing ∗ for ∗ρ, and Q[t], Q≤N [t] ∈ H and γ(Q) in placeof either φc[t] ∗ f, φc;≤N [t] ∗ f and γ(φ ∗ f) or φ∗c [t] ∗ f, φ∗c;≤N [t] ∗ f and γ(φ∗ ∗ f)

respectively. In this section we allow Σ+ to be non-compact, however it shouldbe noted that a convergence of the twisted measure for Q[t] = (φc;≤N [t])∗ ∗ f willultimately only be verified under the assumption that Σ+ is compact.


5.1. Weighted Dirac mass construction.

Definition 5.1. (Approximating measures) Assume that the slowly increasing func-tion c : N→ R has

limN→∞

〈φc;≤N [t] ∗ f,Q≤N [t]〉 → ∞ as t→ γ(Q)

and

limN→∞

〈f,Q[t]〉〈φc;≤N [t] ∗ f,Q≤N [t]〉

→ 0 as t→ γ(Q).

The normalising factor is F (Q)[t] = limN→∞〈φc;≤N [t] ∗ f,Q≤N [t]〉. Formallydefine, for t > γ(Q) and N ∈ N,

νQc;N [t] =∑B∈W1

N∑n=1

t−ncn∑

v∈WB,an

Rn(vx)〈ρ(s(v))f,Q≤N [t]〉

F (Q)[t]D(vx),

in which D(z) is the Dirac mass at z ∈ Σ+, and x ∈ Σ+ is chosen with x ∈ [A] andσx not containing A (this choice relates to a remainder term in Proposition 5.6 andrelates to the choice of first returns in the proof of Theorem 6.6).

For technical reasons we will also make reference to the following family of mea-sures, defined with respect to a group element g ∈ G and slowly increasing functiond : N→ R+. Formally define, for t > γ(Q) and N ∈ N,

(5.1) νg−1Qd;N [t] =

∑B∈W1

N∑n=1

t−ndn∑

v∈WB,an

Rn(vx)〈ρ(s(v))f, ρ(g)−1Q≤N [t]〉

F (Q)[t]D(vx)

for the same x ∈ Σ+ as in Definition 5.1. (Recall that Q is defined in terms of theslowly increasing function c.)

Remark 5.2. For Q≤N [t] = φc;≤N [t] ∗ f let us remark that the well definedness

of νQc;N [t] follows in the same way as checking for νN [t] given in equation 5.3. Theproof is subsumed in Theorem 6.6.

One could check whether νQc;N [t] is well-defined for Q≤N [t] = φ∗c;≤N [t] ∗ f butultimately we have no method to verify whether a limit in N is well-defined, unlessΣ+ is compact

Lemma 5.3. Let g ∈ G. For each B we have that the linear functional νgQc;N [t] has

supt>γ(Q)

supN∈N

νgQc;N [t]([B]) <∞.

If Ts is transitive then

lim inft→γ(Q)

supN∈N

νgQc;N [t]([B]) = lim inft→γ(Q)

limN→∞

νgQc;N [t]([B]) > 0.

In general we have

supN∈N

νQc;N [t]([A]) = limN→∞

νQc;N [t]([A]) = 1

for all t > γ(Q).

If the family νQc;N [t] is tight in N ∈ N, for a fixed t, then νQc;N [t] converge to alimit measure, for a fixed t.

Proof. It is immediate that

νQc;N [t]([B]) =〈φB,ac;≤N [t] ∗ f,Q≤N [t]〉

F (Q)[t].

18 RHIANNON DOUGALL

If Q = φc ∗ f then we use Lemma 2.4 and the Cauchy-Schwarz inequality to get theupper bound

νQc;N [t]([B]) ≤ ‖φB,ac [t] ∗ f‖‖Qc[t]‖

F (Q)[t]≤ Const.(B),

using Lemma 2.6 in the last inequality. If Qc = φ∗c ∗ f then we use Lemma 2.6 toget the upper bound

νQc;N [t]([B]) ≤ Const.(B)〈φA,ac;≤N+KB

[t] ∗ f,Q≤N [t]〉F (Q)[t]

≤ Const.(B)F (Q)[t]

F (Q)[t].

The lower bound is seen using

νQc;N [t]([B]) =〈φB,ac;≤N [t] ∗ f,Q≤N [t]〉

F (Q)[t]

≥ Const.(B)−1〈φA,ac;≤N−KB [t] ∗ f,Q≤N [t]〉

F (Q)[t]

≥ Const.(B)−1〈φA,ac;≤N [t] ∗ f,Q≤N [t]〉

F (Q)[t]− Const.(B)−1

〈φA,ac;≤KB [t] ∗ f,Q≤N 〉F (Q)[t]

.

We check that the linear functionals νg−1Qc;N [t] converge as N → ∞. (We check

that they are Cauchy.) In the first case Q = φc ∗ f . For arbitrary N,M we have

νg−1Qc;N [t]([B])− νg

−1Qc;M [t]([B])

=〈φB,ac;≤N [t] ∗ f − φB,ac;≤M [t] ∗ f, ρ(g)−1φc[t] ∗ f〉

‖φ[t]c ∗ f‖2

≤‖φB,ac;≤N [t] ∗ f − φB,ac;≤M [t] ∗ f‖

‖φ[t]c ∗ f‖.

Strong convergence (Lemma 2.4) gives that ‖φB,ac;≤N [t] ∗ f − φB,ac;≤M [t] ∗ f‖ ≤ ε forN,M sufficiently large.

In the second case Q = φ∗c ∗ f . For each g we use Lemma 2.6 to show that

νg−1Qc;N [t]([B])− νg

−1Qc;M [t]([B])

≤ Const.(B)〈φc;≤N+KB [t] ∗ f,Q≤N [t]〉

F (Q)[t]− Const.(B)−1 〈φc;≤M−KB [t] ∗ f,Q≤M [t]〉

F (Q)[t]

The result follows upon verifying the claim that for a fixed t > γ(Q) we have

(5.2)〈φc;N−r≤n≤N [t] ∗ f, φ∗c;≤N [t] ∗ f〉

F (Q)[t]→ 0 as N →∞.

Were false, there would be a sequence (Nj)j∈N with

〈φc;Nj−r≤n≤Nj [t] ∗ f, φ∗c;≤Nj [t] ∗ f〉F (Q)[t]

> C,

for some constant C > 0. In particular for J > j,

〈φc;Nj−r≤n≤Nj [t] ∗ f, φ∗c;≤NJ [t] ∗ f〉F (Q)[t]

> C.

Now assuming Nj −Nj−1 > r we would conclude

〈φc;n≤NJ [t] ∗ f, φ∗c;≤NJ [t] ∗ f〉F (Q)[t]

≥J∑j=1

〈φc;Nj−r≤n≤Nj [t] ∗ f, φ∗c;≤NJ [t] ∗ f〉F (Q)[t]

> JC.

This cannot hold for arbitrarily many J whilst t is fixed.


Now if the family of νg−1Qc;N [t] is tight in N we deduce that the limit linear func-

tional is a measure. (See corollary 6.7.) �

We give the definition of a twisted measure conditional on the existence of limits

of νg−1Qc;N [t].

Definition 5.4. Let (tq)q∈N be a sequence with limq→∞ tq = γ(Q) and for which

νg−1Qc [tq] (exists and) converges as q →∞ for every g. We call νφc the limit of νφc [tq]

as q → ∞ a Q∗ ∗ φ one-sided (twisted) measure. We also use the notation νgφc forthe limit of νgφc [tq] as q →∞.

The outcome of subsection 6.2 is to show that limit points exist (under certainhypotheses). The outcome of section 7 is to check that these measure agree withthe notion of twisted in Definition 7.1. Before proceeding any further we show howto obtain the eigenmeasure ν in this way. Define, for t > 1,

(5.3) νN [t] =∑B∈W1

N∑n=1

t−n∑

v∈WB,an

Rn(vx)1

ζ[t]D(vx),

recalling that ζ[t] = ζA,a[t]. In due course we will check that these measure arewell-defined, and show the existence of an accumulation point.

Lemma 5.5. Suppose that ν[t] is a weak* limit of νN [t] as N∞ and ν is a weak*limit νN [tk] as k →∞. We have

L∗ν = ν.

Proof. Let w = w1 · · ·wr be arbitrary. First observe that

L(1[w])(z) =∑

u∈W1:τ(u,x0)=1

R(uz)1[w1w2···wr](uz) = R(w1z)1[w2···wr](z)

and so L(1[w])(z) is bounded and continuous. We have, for any presumed limitalong tk → 1 as k →∞,

ν(L(1[w])

)= limk→∞

limN→∞

∑B∈W1

N∑n=1

t−nk

∑v∈WB,a

n

Rn(vx)1

ζ[t]R(w1vx)1[w2···wn](vx)

= limk→∞

∞∑n=1

t−nk

∑v∈Ww2,a

n

Rn(vx)1

ζ[t]R(w1vx)1[w2···wn](vx)

= limk→∞

∑b∈W1

∞∑n=1

t−nk

∑u∈Wb,a

n

Rn+1(w1 · · ·wnux)1

ζ[t]

= limk→∞

limN→∞

∑b∈W1

N+1∑n=2

t1kt−(n+1)

∑u∈Wb,a

n

Rn+1(w1 · · ·wnux)1

ζ[t]

= ν(1[w])

using the fact that ζ[t] diverges to obtain the last equality. �

5.2. Basic estimates for the approximating measures.

Proposition 5.6 (The main equality). Let wB ∈ W with |w| = r. For any k ≥ rwe have

νg−1Qc;N [t](trR−1

r 1[wB]) =

N−r∑n=1

t−ncncn+r

cn

∑v∈WB,a

n

〈ρ(s(w))ρ(s(v))f, ρ(g)−1Qc;≤N [t]〉〈φc;≤N [t] ∗ f,Qc;≤N [t]〉

Rn(vx)

+

r∑n=1

t−ncn∑b∈W1

∑v∈Wb,a

n

Rn(vx)〈ρ(s(v))f,Qc;≤N [t]〉〈φc;≤N [t] ∗ f,Qc;≤N [t]〉

tr1[wB](vx)

Rr(vx)

20 RHIANNON DOUGALL

Proof. Write wB = duB where d ∈ W1. We have

νg−1Qc;N [t](trR−1

r 1[dub])

=

N∑n=r+1

t−ncn∑

v1···vrv∈Wd,an

Rn(v1 · · · vrvx)〈ρ(s(v1 · · · vrv))f,Qc;≤N [t]〉〈φc;≤N [t] ∗ f,Qc;≤N [t]〉

tr1[dub](v1 · · · vrvx)

Rr(v1 · · · vrvx)

+

r∑n=1

t−ncn∑

v∈Wd,an

Rn(vx)〈ρ(s(v))f,Qc;≤N [t]〉〈φc;≤N [t] ∗ f,Qc;≤N [t]〉

tr1[dub](vx)

Rr(vx)

andN∑

n=r+1

t−ncn∑

v1···vrv∈Wd,an

Rn(v1 · · · vrvx)〈ρ(s(v1 · · · vrv))f,Qc;≤N [t]〉〈φc;≤N [t] ∗ f,Q≤N [t]〉

tr1[dub](v1 · · · vrvx)

Rr(v1 · · · vrvx)

=

N−r∑n=k−r+1

t−ncncn+r

cn

∑v∈WB,a

n

〈ρ(s(du))ρ(s(v))f, ρ(g)−1Qc;≤N [t]〉〈φc;≤N [t] ∗ f,Q≤N [t]〉

Rn(.vx)Rr(duvx)

Rr(.duvx).

�

In Definition 12.5 we gave a local Gibbs definition which is satisfied by an Rconformal measure. Eventually Lemma 8.1 will give a RHS local Gibbs inequal-ity for limits of the twisted measure when Q = φc ∗ f . In order to even checkthat the sequence of approximating measures is tight we will need to check someapproximation of the RHS local Gibbs inequality.

Lemma 5.7 (c-heavy RHS local Gibbs when Q = φc ∗f). Suppose that Q = φc ∗f .For each B there is a constant C(B) so that for any BwA ∈ W we have

νg−1Qc;N [t]([wA]) ≤ Rr(wx)t−rC(B) sup

n∈N

(cn+r

cn

),

where r = |w| and g ∈ G is arbitrary.

Proof. Write γ = γ(Q). Strong convergence tells us that F (Q)[t] = ‖φc[t] ∗ f‖2.

Recall that Proposition 5.6 gives an expression for the term νg−1Qc;N [t](γrR−1

r 1[wA]).As we chose σx to not contain A this forces that the second series is 0 except atn = r from which

r∑n=1

t−ncn∑b∈W1

∑v∈Wb,a

n

Rn(vx)〈ρ(s(v))f, φc;≤N [t] ∗ f〉

‖φc[t] ∗ f‖2tr1[wA](vx)

Rr(vx)

=〈ρ(s(w))f, φc;≤N [t] ∗ f〉

‖φc[t] ∗ f‖2≤ 1.

We give an upper bound for the first series in Proposition 5.6 by(supn≥r

cn+r

cn

) N∑n=1

t−ncn∑

v∈WA,an

〈ρ(s(w))ρ(s(v))f, ρ(g)−1φc;≤N [t] ∗ f〉‖φc[t] ∗ f‖2

Rn(vx)

=

(supn≥k

cn+r

cn

) 〈ρ(s(w))φB,ac;≤N [t] ∗ f, φc;≤N [t] ∗ f〉‖φc[t] ∗ f‖2

Rn(vx)

≤(

supn≥k

cn+r

cn

)‖φB,ac [t] ∗ f‖‖φc[t] ∗ f‖

≤(

supn≥k

cn+r

cn

)Const.(B).

Using local Holder continuity and recalling that we fixed x ∈ [A] gives

νg−1Qc;N [t](γrR−1

r 1[wA]) ≥ const.(wA)γrR−1r (wx)νg

−1Qc;N [t]([wA]).

Recalling const.(wA) does not depend on w, the Lemma follows. �


6. One-sided (twisted) measure: existence

In this section we collect the machinery to show existence of the one-sided(twisted) measures. In particular we must check existence of the slowly increas-ing functions, and will verify the existence of accumulation points by checking thatthe family of approximating measures are tight.

6.1. The slowly increasing function. The assumption that R is recurrent (andconsequent divergence of the return series 12.11) is useful in the construction ofthe R-conformal measure (Lemma 5.5). We can always force a series to divergeat its radius of convergence by increasing the summands with a slowly increasingfunction. This observation is used widely in the literature on conformal measures(for instance [22], [24], [6]). The result is stated as Proposition 15.1.

Lemma 6.1. There is a slowly increasing c with∫〈ρ(h)f, f〉dφ∗c [t] ∗ φc[t] <∞ for t > γ(φ ∗ρ f)

and ∫〈ρ(h)f, f〉dφ∗c [t] ∗ φc[t]→∞ as t→ γ(φ ∗ρ f).

Proof. In particular the Lemma asks us to check that, for any slowly increasing cwe have∫

〈ρ(h)f, f〉dφ∗c [t] ∗ φc[t] <∞ ⇐⇒∫〈ρ(h)f, f〉dφ∗c [t] ∗ φc[t] <∞.

This is seen immediately from the following Since c is slowly increasing, for eachδ < 1 there is Cδ with

〈φc;≤N [t] ∗ f, φc;≤N [t] ∗ f〉 ≤∑m≤N

∑n≤N

Cδ〈φ[δ−1t] ∗ f, φ[δ−1t] ∗ f〉

≤ Cδ〈φ[δ−1t] ∗ f, φ[δ−1t] ∗ f〉 <∞.

Now we check that γ := γ(φ ∗ρ f) is the abscissa of convergence of the real series

π[t] =

∞∑N=1

t−N∑

m+n=N

∑V ∈WA,a

m

∑v∈WA,a

n

Rm(V x)Rn(vx)〈ρ(s(V ))−1ρ(s(v))f, f〉.

Write

π≤N [t] =∑K≤N

t−K∑

m+n=K

∑V ∈WA,a

m

∑v∈WA,a

n

Rm(V x)Rn(vx)〈ρ(s(V ))−1ρ(s(v))f, f〉

Set φn[t](g) =∑w∈WA,a

n :s(w)=g Rn(wx). It is easy to see that

π≤N [t] ≤N−1∑n=1

N−1∑m=1

〈φn[t] ∗ f, φm[t] ∗ f〉 = 〈φ≤N [t] ∗ f, φ≤N [t] ∗ f〉

On the other hand

〈φ≤N [t] ∗ f, φ≤N [t] ∗ f〉 =∑k≤2N

∑m+n=k:

m≤N,n≤N

〈φn[t] ∗ f, φm[t] ∗ f〉

≤∑k≤2N

∑m+n=k

〈φn[t] ∗ f, φm[t] ∗ f〉 = π≤2N [t]

It follows that the abscissa of convergence of π[t] is γ.Using Proposition 15.1, choose dN slowly increasing so that

πd[t] =

∞∑N=1

t−NdN∑

m+n=N

∑V ∈WA,a

m

∑v∈WA,a

n


22 RHIANNON DOUGALL

diverges at γ. Now write cn = d2n, and observe that c inherits the slowly increasingproperty from d. Then for m+ n = N we have cmcn = d2md2n ≥ dN , giving

πd≤N [t] =∑K≤N

t−KdK∑

m+n=K

∑V ∈WA,a

m

∑v∈WA,a

n


≤∑K≤N

t−K∑

m+n=K

cmcn∑

V ∈WA,am

∑v∈WA,a

n


≤ 〈φc[t] ∗ f, φc[t] ∗ f〉.

The conclusion follows. �

The divergence statement for φ∗ involves more attention. We begin by verifyingthat a slowly increasing function does not increase the convergence parameter.

Lemma 6.2. Let c : N→ R+ be subexponentially increasing. Then

supN∈N〈φc;≤N [t] ∗ f, f〉 <∞ ⇐⇒ sup

N∈N〈φ≤N [t] ∗ f, f〉 <∞.

If c is slowly increasing then

supN∈N〈φc;≤N [t] ∗ f, φ∗c;≤N [t] ∗ f〉 <∞ ⇐⇒ sup

N∈N〈φ≤N [t] ∗ f, φ∗≤N [t]f〉 <∞.

If Ts is transitive then γ(φ∗ ∗ f) = γ(f).

Proof. Fix f . To begin with we only ask that c : N → R+ has subexponentialgrowth lim supn→∞

1n log cn = 0. (If cn is slowly increasing then log cn − log ck ≤

log cncn−1

+ · · · log ck+1

ck≤ (n− k) log γ for γ arbitrarily close to 1 and k = kγ .)

We have ∫〈ρ(g)f, f〉dφ≤N [t](g) = 〈φc;≤N [t] ∗ρ f, f〉 =: η≤N [t],

and ∫〈ρ(g)f, f〉dφc;≤N [t](g) = 〈φc;≤N [t] ∗ρ f, f〉 =: ηc;≤N [t].

Note ηc[t] = limN→∞ ηc;≤N [t], η[t] = limN→∞ η≤N [t] are real power series in t−1.For each δ < 1 there is Cδ with ηc[t] ≤ Cδηc[δ−1t]. The first statement follows.

Now set

〈φ≤N [t] ∗ φ≤N [t] ∗ρ f, f〉 =: α≤N [t],

and

〈φc;≤N [t] ∗ φc;≤N [t] ∗ρ f, f〉 =: αc;≤N [t].

To be clear αc[t] = limN→∞ αc;≤N [t] does not have form of a real power series int−1, it is

αc[t] = limN→∞

∞∑K=1

t−K∑

m+n=K,m≤N,n≤N

∑v∈WA,a

n

∑u∈WA,a

m

Rn(ux)Rn(vx)〈ρ(s(u))ρ(s(v))f, f〉.

However setting

πc;≤N [t] =

N∑K=1

t−K∑

m+n=K

cncm∑

v∈WA,an

∑u∈WA,a

m

Rn(ux)Rn(vx)〈ρ(s(u))−1ρ(s(v))f, f〉

we have that πc = limN→∞ πc;≤N [t] is a real power series in t−1, and we check that

πc;≤N [t] ≤ αc;≤N [t] ≤ πc;≤2N [t].

Assuming that Aa are admissible we have

(6.1) πc;≤2N [t] ≤ ηc∗c;≤2N [t].


By assumption cn is slowly increasing, from which we deduce that log(c ∗ c)n ≤log n+ log cn and so c ∗ c grows subexponentially. Using the first of the lemma wededuce that for t > γ(φ ∗ f) we have αc[t] is finite.

We conclude by mentioning that

η[t] ≤ Cαc[t]follows when Ts is transitive. This tells us that γ(φ∗ ∗ f) = γ(f). �

We are now ready to prove the divergence statements. Let us mention separatelythe case f = δe which follows easily.

Proposition 6.3. Assume that Ts is transitive. There is c with

limN→∞

〈φc;≤N [t] ∗ δe, φ∗c;≤N [t] ∗ δe〉 → ∞ as t→ γ(δe)

and

limN→∞

〈φc[t] ∗ δe, δe〉〈φc;≤N [t] ∗ δe, φ∗c;≤N [t] ∗ δe〉

→ 0 as t→ γ(δe).

Proof. Note that the transitivity hypothesis ensures that γ(φ∗ ∗ δe) = γ(δe).Choose c to be a slowly increasing function with 〈φc[t] ∗ δe, δe〉 → ∞ as t → γ

(recall that η[t] in the proof of Lemma 6.2 is a real power series in t−1). Thenimmediately we have limN→∞〈φc;≤N [t] ∗ φc;≤N [t] ∗ δe, δe〉 → ∞ as t→ γ. We havethat

〈φc;≤N [t] ∗ δe, φ∗c;≤N [t] ∗ δe〉 ≥ 〈φc;≤N [t] ∗ δe, δe〉〈δe, φ∗c;≤N [t] ∗ δe〉.It follows that

limN→∞

〈φc[t] ∗ δe, δe〉〈φc;≤N [t] ∗ δe, φ∗c;≤N [t] ∗ δe〉

= limN→∞

〈φc[t] ∗ δe, δe〉〈φc;≤N [t] ∗ δe, δe〉2

→ 0 as t→ γ(φ∗ ∗δe).

�

In general we use strong positive recurrence to check that the convolution φc;≤N [t]∗φc;≤N [t] “is bigger” than φc;≤N [t]. In order to do this we need to use the secondoutcome of Proposition 15.1 that says that we can choose c with cn+k ≤ cnck forall n, k ∈ N. This helps us estimate the convolution of c with itself.

Lemma 6.4. Assume that Ts is transitive. There is c with

limN→∞

〈φc;≤N [t] ∗ f, φ∗c;≤N [t] ∗ f〉 → ∞ as t→ γ(f)

and

limN→∞

〈φc[t] ∗ f, f〉〈φc;≤N [t] ∗ f, φ∗c;≤N [t] ∗ f〉

→ 0 as t→ γ(f).

Proof. The number of ways a word w ∈ WA,am can be written as w = uv, for words

in u ∈ WA,an , v ∈ WA,a

k with n+ k = m, depends on the number of times an orbitin w returns to Aa. We have

φc;≤N [t] ∗ φc;≤N [t] ≥ C−1N∑m=1

t−m∑

w∈WA,am

Rm(wx)δs(w)

∑j≤N :wm−jwm−j−1=Aa

cjcm−j

≥ C−1N∑m=1

t−mcm∑

w∈WA,am

Rm(wx)δs(w)# {j ≤ N : wm−jwm−j−1 = Aa} .

Fix M . Consider WA,am (< M) those words with # {j ≤ N : wm−jwm−j−1 = Aa} <

M . By strong positive recurrence we have

N∑m=1

cmt−m

∑w∈WA,a

m (<M)

Rm(wx) = AM (t)

24 RHIANNON DOUGALL

converges for t ∈ (γ(SPR) + ε,∞) and in particular supt∈[γ(f),γ(f)+1]AM (t) < ∞.We also note that

N∑m=1

t−mcm∑

w∈WA,am (<M)

Rm(wx)〈δs(w) ∗ f, f〉 ≤ AM (t).

It follows that

〈φc;≤N [t] ∗ φc;≤N [t] ∗ f, f〉 ≥ C−1M〈φc;≤N [t] ∗ f, f〉 − C−1AM (t),

i.e.

lim supN→∞

〈φc;≤N [t] ∗ φc;≤N [t] ∗ f, f〉〈φc[t] ∗ f, f〉

≥ C−1M − C−1AM (t)

〈φc[t] ∗ f, f〉.

Using the divergence of 〈φc[t] ∗ f, f〉 gives the conclusion. �

Remark 6.5. If d, d : N → R have that limn→∞ dn/cn = 1 = limn→∞ dn/cn thenfor the c in Lemma 6.4 we also have

limN→∞

〈φc[t] ∗ f, f〉〈φd;≤N [t] ∗ f, φ∗d;≤N [t] ∗ f〉

→ 0 as t→ γ(f).

6.2. Tightness results. If Σ+ is compact then any collection of measures withbounded mass is tight, and by Lemma 5.3 we know this for Q = φ∗ ∗ f and νgφ

∗

c [t].We now let Σ+ be a countable Markov shift and assume that R is strongly

positively recurrent. We can only handle the case Q = φ ∗ f .

Theorem 6.6. Assume that Q[t] = φ[t] ∗ f . Assume that

t 7→∞∑r=1

t−r supn∈N

(cn+r

cr

) ∑w∈WA,a(∗)

Rr(wx)

converges at t = γ(Q). For every B and for every ε > 0 there is a compact set Kfor which

νg−1Qc;N [t]((Σ+ −K) ∩ [B]) ≤ ε

andνN [t′]((Σ+ −K) ∩ [B]) ≤ ε

for every N ∈ N, g ∈ G and t > γ(Q), t′ > 1.

Corollary 6.7. Assume that γ(Q) > γ(SPR) and that c is subexponentially in-

creasing. For each t > γ(Q) there are measures νg−1Qc [t] on Σ+, finite on cylinders,

with νg−1Qc;N [t] → νg

−1Qc [t] as N → ∞ in the weak* topoology. There is a sequence

tk → γ(Q) and measures νg−1Qc on Σ+, finite on cylinders, with νg

−1Qc [tk]→ νg

−1Qc

as k →∞ in the weak* topology. In addition νQc ([A]) = 1.

Proof. The hypotheses to the corollary imply that for β < 1∞∑r=1

t−r supn∈N

(cn+r

cr

) ∑w∈WA,a(∗)

Rr(wx) ≤ C∞∑r=1

(βt)−r∑

w∈WA,a(∗)

Rr(wx)

which coverges for βγ(Q) > γ(SPR).One does have to be careful about extracting accumulation points when Σ+ is

non-compact — it is the tightness result of Theorem 6.6 that implies the existenceof accumulation points. Let us sketch of the details of this well-known mechanism.Suppose mk are tight in k ∈ N. For a fixed B and compact set Kn in the tightnesscriterion, we have that the measures mk restricted to Kn∩[B] have an accumulationpoint that is a positive measure. So, for some subsequence mkn(q) converge asq →∞ to a positive measure. Using nesting of Kn we may assume that kn(q) is asubsequence of kn−1(q). Then setting qn = kn(n) we have that mqn converges forany Kn, and in particular the limit measure is well-defined on the union, which is


[B]. We use a similar diagonal sequence have convergence along a sequence whichworks for all B. We use a similar diagonal sequence to deduce convergence along asequence tk → γ(Q) which works for all g. To see that the limit is finite on cylinderswe using the first part of Lemma 5.3 which tells us that the measure of a cylinderis bounded from above in t. �

We proceed in a similar fashion as Sarig [20]. In our case we rely on Lemma 5.7to estimate the measure of cylinders.

Without loss of generality we may assume that Σ+ ⊂ NN; that is we represent thetransitions by natural numbers. Let us assume that A is represented by 1 ∈ N. Weneed notation governing the first returns. Let B ∈ W1. For a sequence of numbers(Mp)p∈N set:

• K((Mp)) = {x ∈ Σ : ∀p ∈ Nxp ≤Mp} .• Bn = {w : w = w1 · · ·wn−1A ∈ Wn, τ(B,w1) = 1, wi > 1, i = 1, · · · , n− 1}• An = {w : w = w1 · · ·wn−1A ∈ Wn, τ(A,w1) = 1, wi > 1, i = 1, · · · , n− 1};• A =

⋃∞n=1An;

• for k ∈ N, An(≥Mk) = {w : w = w1 · · ·wn ∈ An∃i = 1, · · · , nwi > Mk};• A(≥Mk) = {w : w = w1 · · ·wn ∈ An, n ∈ N,∃i = 1, · · · , nwi > Mk};• Aj is the usual cartesian product.

We write c : N→ Rcr = sup

n∈N

(cn+r

cr

)We write

ηA[t] :=

∞∑n=1

t−ncn∑w∈An

Rn(Aw1 · · ·wn−1x),

and

ξB [t] :=

∞∑n=1

t−ncn∑w∈Bn

Rn(Bw1 · · ·wn−1x),

recalling that x ∈ [A] (see Definition 5.1).The set K((Mp)) is compact. And

Σ+ −K((Mp)) = {x ∈ Σ : ∃q ∈ Nxq > Mq} .We will always assume that Mk ≤Mk+1.

For brevity we write mN [t] = νg−1Qc;N [t]. The only tool we use is Lemma 5.7 which

applies uniformly in g.

Claim 6.8. Assuming Mp are large enough, if z ∈ supp(mN [t]) and z ∈ (Σ −K((Mp))) ∩ [B], it must be that either

1 z ∈ [u], with u ∈ B(≥M1), orj z ∈ [uvw] with u ∈ B, v ∈ Aj, and w ∈ A(≥Mj), for some j ∈ N.

Recall the constant C(B) appearing in Lemma 5.7, which we may assume exceedsthe local Holder constant.

Claim 6.9. We have

1: For u ∈ B(≥M1), |u| = r + 1,

mN [t]([u]) ≤ C(B)t−r crRr(ux)

j: For u ∈ B(≥ M1), |u| = r + 1, v ∈ Aj, |v| = p, w ∈ Ak(≥ Mj), for somej ∈ N;

mN [t]([uvw])

≤ C(B)3t−k ckRk(Awx)t−r crRr(ux)t−pcpRp(Avx)

We use that C(B) exceeds the local Holder constant and that cm+k ≤ ck cm.

26 RHIANNON DOUGALL

Claim 6.10. We have

1:

mN [t]({[u] : u ∈ Bk(≥M1), k ∈ N})

≤ C(B)2∑k∈N

tk−1ck−1

∑u∈Bk(≥M1)

Rk−1(ux)

j:

mN [t]({

[uvw] : v ∈ Aj , u ∈ B, w ∈ Ak(≥Mj), j, k ∈ N}

)

≤ C(B)2ξB(t)(C(B)ηA[t]

)2j+1∑k∈N

tk ck∑

w∈Ak(≥Mj)

Rk(Awx)

Claim 6.11. For every λ > 0 there is choice of Kλ (uniform in t > γ(Q)) so that∑k∈N

tk ck∑

w∈Ak(≥Kλ)

Rk(wx) ≤ λ

and ∑k∈N

tk ck∑

w∈Bk+1(≥Kλ)

Rk(wx) ≤ λ.

Proof of Theorem 6.6. Let ε > 0. For q ∈ N, set

λq =ε

2q(C(B))−1

(C(B)ηA[t]

)−2q−1

and let Kλq be given as in the claim. Set Mq = Kλq . Then

mN [t](Σ−K((Mp)))

≤∑q∈N

C(B)(C(B)ηA[t]

)2q+1∑k∈N

tk ck∑

w∈Ak(≥Mq)

Rk(wx)

≤ ε∑q∈N

1

2q≤ ε

�

7. Twist by cocycle, and (limit of) matrix coefficients

Section 6 verifies cases where the twisted measures (as in Definition 5.4) exist. Inthis section we continue to elaborate on their propertes. First we give a digressioninto the terminology of twisted measures. Let us recall that the construction of ourtwisted measures originates (albeit in a different form) in [5], where the twisted mea-sure is operator valued. In [5] one may understand a “twisting” phenomena takingplace in the operator space, whereas here we discuss a real-valued counterpart.

7.1. Local branches, multiplicative cocycles, and twisted measures. Themap σ : Σ+ → Σ+ is not invertible but on any cylinder [w] with |w| = n the localbranch σ(w) : [w]→ σn[w], σ(w)(z) = σn(z) has a local (left) inverse τ (w) : σn[w]→[w]. The measures (τ (w))∗m, (σ(w))∗m have the defining property∫F (τ (w))∗m =

∫F◦τ (w)(z)1σn[w]dm(z),

∫F (τ (w))∗m =

∫F◦σ(w)(z)1[w]dm(z).

Choosing F = 1[w] we have

(τ (w))∗m([w]) = m(σn[w]), (σ(w))∗m([w]) = m([ww]),

if ww is admissible. The measure (τ (w))∗m that is supported in [w] and so it makessense to ask whether it is absolutely continuous to m restricted to [w]; in this way

the Radon-Nikdoym derivatived(τ(w))∗m

dm (z) is only defined in [w]. Whereas the


measure (σ(w))∗m that is supported in σn[w] and the Radon-Nikdoym derivatived(σ(w))∗m

dm (z) is defined in σn[w]. It can be checked that

d(σ(w))∗m

dm(z) =

(d(τ (w))∗m

dm(τ (w)z)

)−1

In general we have no reason to be able to extend τ (w) and σ(w) to a groupaction (compare with section 14). We are, however, able to generalise the cocycleRadon-Nikodym derivate aspect of the group action using the structure of a groupextension s :W1 → G. We introduce some terminology.

Definition 7.1. Let m be a probability measure on Σ+. If there exist γ > 0 andh : G× Σ+ → R satisfying

(7.1)d(τ (w))∗m

dm(z) = γnRn(z)−1h(s(w), z),

for every w ∈ Wn and z ∈ [w] then we say that m is twisted by h. We call such anh a generalised multiplicative cocycle.

In [5] the word “twisted” can be thought of as referring to a unitary twist. Herewe use the term “twist” to mean twisted by h. We will check that the measurem = νcQ is twisted in the sense of Definition 7.1, for Q = φ ∗ f, φ∗ ∗ f .

Let us conclude the digression with the following. We have the identity

h(e, z) = h(g−1, z)h(s(w), τ (w)z)

whenever g = s(w) and z ∈ σn[w]. This is of interest because, on the one hand forg = s(w), |w| = n,∫σn[w]

h(g−1, z)

h(e, z)dm(z)

∫σn[w]

1

h(g, τ (w)z)dm(z) =

∫1[w](τ

(w)z)1

h(s(w), τ (w)z)dm(z)

=

∫1[w](z)

1

h(s(w), z)d(τ (w))∗m(z)

=

∫1[w](z)

1

h(s(w), z)

d(τ (w))∗m

dm(z)dm(z)

=

∫1[w](z)γ

nR−1n (z)dm(z).

And on the other hand when m = νcφ∗f we evaluate the final term in terms of (limits

of) matrix coefficients at g.

7.2. Technical lemmas.

Lemma 7.2. Assume γ(Q) > γ(SPR). Then νg−1Qd;N [t] converge to a measure, finite

on cylinders, as N →∞. In this way

νg−1Qd [t] =

∑b∈W

∞∑n=1

t−ndn∑

v∈Wb,an

Rn(vx)〈ρ(s(v))f, ρ(g)−1Q[t]〉

F (Q)[t]D(vx)

is well defined, and for each B the νQd;N [t] measure of [B] is bounded uniformly in

t > γ(Q).

Proof. Let C = max(sup{dkck

: ck 6= 0}, sup

{ckdk

: dk 6= 0}

). We can transfer any

cylinder bounds for νQc;N [t] to νQd;N [t] since C−1νQd;N [t](E) ≤ νQc;N [t](E) ≤ CνQd;N [t](E)for every open set. �

28 RHIANNON DOUGALL

“The main equality” of Proposition 5.6 can be interpreted as saying that∫FdtkR−1

k νQc;N [t] = νg−1Qd;N [t](F ) + rem.g

−1Qd;N ;[1,r][t](t

kR−1k F ) + rem.g

−1Qd;N ;[N−r,N ][t](F )

with dn = 1[r,∞)(n)cn+r and rem.g−1Qd;N ;I [t] the measure

rem.g−1Qd;N ;I [t] =

∑n∈I

dnt−n

∑v∈WB,a

n

〈ρ(s(v))f, ρ(g)−1Q≤N [t]〉F (Q)[t]

.

Proposition 7.3. For any k

νg−1Qd;N [t] = νg

−1Qd;N [t] + rem.g

−1Qd,N ;[1,k][t]

with dn = 1[k,∞)(n)cn

We first check that the two remainder terms to go zero.

Lemma 7.4. For each g, r, and d, we have

rem.g−1Qd,N ;[1,r][t]→ 0 as t→ γ.

Proof. We have

rem.g−1Qd,N ;[1,r][t]([B]) =

〈φB,ad;≤r ∗ f, ρ(g)−1Q≤N [t]〉F (Q)[t]

.

First note that d/c is bounded in the range [1, r] so

rem.g−1Qd,N ;[1,r][t]([B]) ≤ C

〈φB,av;≤r ∗ f, ρ(g)−1Q≤N [t]〉F (Q)[t]

,

for some C > 0. Second, use Lemma 2.4 to show that

rem.g−1Qd,N ;[1,r][t]([B]) ≤ Const.(B)C

〈ρ(hB)φA,ac;≤r+KB ∗ f, ρ(g)−1Q≤N [t]〉F (Q)[t]

,

for some hB .If Ts is transitive we may assume hB = g−1 and then use that

〈φA,ac;≤r+KB ∗ f,Q[t]〉F (Q)[t]

→ 0 as t→ γ,

using Lemma 6.4 and that r,KB are fixed.If Q = φc ∗ f then we use the Cauchy-Schwarz inequality and then see that

‖φA,ac;≤r+KB ∗ f‖‖Q≤N [t]‖F (Q)[t]

→ 0 as t→ γ,

using Lemma 6.1 and that r,KB are fixed. �

Lemma 7.5. For each g, r and d with d/c bounded, we have

rem.g−1Qd,N ;[N−r,N ][t]→ 0 as N →∞.

Proof. We have

rem.g−1Qd,N ;[N−r,N ][t]([B]) =

〈φB,ad;N−r≤n≤N ∗ f, ρ(g)−1Q≤N [t]〉F (Q)[t]

The easier case is Q = φc ∗ f . We have the upper bound

rem.g−1Qd,N ;[N−r,N ][t]([B]) ≤

(sup

n∈[N−r,N ]

dncn

)Const.(B)

〈ρ(hB)φA,ac;N−r≤n≤N+KB[t] ∗ f, ρ(g)−1Q≤N [t]〉

F (Q)[t].


Using Cauchy-Schwarz gives

rem.g−1Qd,N ;[N−r,N ][t]([B]) ≤

(sup

n∈[N−r,N ]

dncn

)Const.(B)

‖φA,ac;N−r≤N+KB∗ f‖‖Q≤N [t]‖

F (Q)[t].

We know that ‖φA,ac;N−r≤N+KB∗f‖ → 0 as N →∞ by strong convergence of φc;≤N ∗f

to φ[t] ∗ f . It follows that rem.g−1Qd,N ;[N−r,N ][t]([B])→ 0 as N →∞.

In the other case Q = φ∗c ∗ f and we assume that Ts is transitive. We obtain

rem.g−1Qd,N ;[N−r,N ][t]([B]) ≤

(sup

n∈[N−r,N ]

dncn

)Const.(B)

〈φA,ac;N−r≤n≤N+KB[t] ∗ f, φ∗c;≤N [t] ∗ f〉

F (Q)[t].

We use 5.1 to conclude. �

Now we check how the limit of νg−1Qd depends (or doesn’t depend) on d.

Lemma 7.6. If cndn→ 1 as n → ∞ then (νg

−1Qd [tk] converge as k → ∞, the limit

measure has) νg−1Qd = νg

−1Qc .

Proof. Let the cylinder [w] be arbitrary. Let ε > 0 be arbitrary. Using the hy-pothesis on d, choose K sufficiently large with cn/dn ≤ 1 + ε for n ≥ K. UsingProposition 7.3 we have

νg−1Qd;N [tq]([w]) = νg

−1Qd;N [tq]([w]) + rem.g

−1Qd,N ;[1,k][t].

Now,

νg−1Qd [tq]([w]) ≤ (1 + ε)νg

−1Qc [tq]([w])

We use Lemma 7.4 to conclude that the remainder term vanishes as q →∞. Sinceε was arbitrary the conclusion follows. �

Remark 7.7. The propositions and lemmas 7.3 7.4 7.5 7.6 are all valid when wereplace ρ with the anti-homomorphism ρ∗(g) = ρ(g)−1. This observation will saveus from almost identical arguments in section 10.

Lemma 7.8. If Ts is transitive then νg−1Qc is absolutely continuous with respect to

νQc .

Proof. It is enough to check that for each w we have

νg−1Qc (R−1

|w|[wB]) ≤ CνQc (R−1|w|[wB]).

There is a constant Kg,Const.(g) with

νg−1Qc [t](R−1

|w|[wB])− rem.g−1Qc,N ;[1,|w|][t] =

〈ρ(s(w))φB,ac;≤N−|w|[t] ∗ f, ρ(g)Qc;≤N [t] ∗ f〉F (Q)[t]

≤ Const.(g)〈ρ(s(w))φB,ac;≤N−|w|[t] ∗ f,Q≤N+Kg [t] ∗ f〉

F (Q)[t]

≤ Const.(g)〈ρ(s(w))φB,ac;≤N+Kg−|w|[t] ∗ f,Q≤N+Kg [t] ∗ f〉

F (Q)[t]

≤ Const.(g)(νQc;N+Kg

(R−1|w|[w])− rem.Qc,N+Kg ;|w|[t]

).

Using the results on remainders (Lemma 7.4) the conclusion follows. �

30 RHIANNON DOUGALL

7.3. The twist and the (limits of) matrix coefficients.

Lemma 7.9. If Ts is transitive then νQc is twisted in the sense of Definition 7.1.

Proof. For brevity write m = νQc , mg = νg−1Qc and γ = γ(Q). The measure mw

defined by ∫Fdmw =

∫Fγ−nRnd(τ (w))∗m

has mw = mg−1

for a certain d and g = s(w). In addition mg is absolutely contin-uous with respect to m by Lemma 7.8. Since mw(F ) = mv(F ) for every w, v withs(w) = s(v) = g it follows that dmw

dm (z) = dmvdm (z) =: h′(g, z), i.e. a function indexed

by the group element. Now using the chain rule for Radon-Nikodym derivatives wehave

h′(g, z) =dmw

dm(z) =

dγ−nRn(τ (w))∗m

d(τ (w))∗m(z)

d(τ (w))∗m

dm(z) = γ−nRn

d(τ (w))∗m

dm(z).

�

It is now easier to write the conclusions for the (limits of) matrix coefficients.

Lemma 7.10. For any wA we have∫[wA]

γ(Q)nR−1n dνQc = lim

q→∞limN→∞

〈ρ(s(w))φc;≤N [tq] ∗ f,Q≤N [tq]〉〈φc;≤N [tq] ∗ f,Q≤N [tq]〉

If Q = φc ∗ f then∫[wA]

γ(Q)nR−1n dνQc = lim

q→∞

〈ρ(s(w))φc[tq] ∗ f, φc[t] ∗ f〉〈φc[tq] ∗ f, φc[tq] ∗ f〉

.

Lemma 7.11. Suppose Ts is transitive. We have∫[wB]

γ(Q)nR−1n dνQc ≤ CB lim

q→∞limN→∞

〈ρ(s(w))φc;≤N [tq] ∗ f,Q≤N [tq]〉〈φc;≤N [tq] ∗ f,Q≤N [tq]〉

.

If Q = φc ∗ f then∫[wB]

γ(Q)nR−1n dνQc ≤ CB lim

q→∞

〈ρ(s(w))φc[tq] ∗ f, φc[t] ∗ f〉〈φc[tq] ∗ f, φc[tq] ∗ f〉

.

Proof of Lemmas 7.10 and 7.11. By the main equality and Lemmas 7.4 and 7.5,

νQc (γ(Q)nR−1n 1[wB]) = νg

−1Qc ([B]),

for g = s(w). We are always using that fact that R−1n 1[wB] is bounded and continu-

ous so that we can deduce its integral with respect to the weak limit νQc . We knowthat

νg−1Qc ([B]) = lim

q→∞limN→∞

〈ρ(g)φB,ac;≤N [tq] ∗ρ f,Q≤N [tq]〉HlimM→∞〈φc;≤M [tq] ∗ρ f,Q≤M [tq] ∗ρ f〉H

,

and since the numerator and denominator have bounded limits we may say

νg−1Qc ([B]) = lim

q→∞limN→∞

〈ρ(g)φB,ac;≤N [tq] ∗ρ f,Q≤N [tq]〉H〈φc;≤N [tq] ∗ρ f,Q≤N [tq] ∗ρ f〉H

.

In the case that Q = φc ∗ f we use strong convergence to say, for B = A,

limN→∞

〈ρ(g)φc;≤N [tq] ∗ρ f, φc;≤N [tq]〉HlimM→∞〈φc;≤M [tq] ∗ρ f, φc;≤M [tq] ∗ρ f〉H

=〈ρ(g)φc[tq] ∗ρ f, φc[tq]〉H〈φc[tq] ∗ρ f, φc[tq] ∗ρ f〉H

.

If Ts is transitive then using Lemma 2.6

νg−1Qc ([B]) ≤ CB lim

q→∞limN→∞

〈ρ(g)φA,ac;≤N [tq] ∗ρ f,Q≤N [tq]〉H〈φc;≤N [tq] ∗ρ f,Q≤N [tq] ∗ρ f〉H

completing the proof. �


7.4. Further estimates. We check that the (limits of) matrix coefficients are non-trivial.

Corollary 7.12. Assume Σ+ compact. Suppose Ts is transitive. For each g wehave

limq→∞

limN→∞

〈ρ(g)φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉〈φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉

> 0.

If γ(Q) < 1 then for every ε > 0 there exist h with

limq→∞

limN→∞

〈ρ(g)φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉〈φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉

< ε.

Proof. Choose w with s(w) = g. Then

lim supq→∞

〈ρ(g)φc[tq] ∗ f, φc[tq] ∗ f〉‖φc[tq] ∗ f‖2

≥ CνQc (1[wA]) ≥ CνQc (1[wAv])

for some constant C and any v. If we choose v with s(wAv) = e then we get positivelower bound for νQc (1[wAv]).

For the second part we have that there is some constant C so that for each n

CνQc (1) ≥∑b∈W1

∑w∈Wb,A

n

γ(Q)−nRn(wx) limq→∞

limN→∞

〈ρ(s(w))φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉〈φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉

.

Since the∑b∈W1

∑w∈Wb,A

nγ(Q)−nRn(wx) diverges in n there must be h with

limq→∞

limN→∞

〈ρ(h)φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉〈φc;≤N [tq] ∗ f,Q≤N [tq] ∗ f〉

arbitrarily small. �

8. The local RHS Gibbs property and absolute continuity at themaximal parameter

These manipulations are only valid for the case νφc∗fc measure. We set γ =γ(φc ∗ f). Using that Υc(g) ≤ 1 we have the following lemma.

Lemma 8.1 (RHS local Gibbs). There is a constant C so that for any word w withadmissible wA and |w| = r we have

νφ∗fc ([wA]) ≤ CRr(wx)γ−r

Theorem 8.2. If γ = 1 then νφc∗fc is absolutely continuous with respect to ν.

Proof. Fix B. Let ε > 0. We must find η > 0 for which νφc∗fc (E ∩ [B]) < ε, for anyopen set E with ν(E ∩ [B]) < η. For brevity we assume always that E ⊂ [B].

Recall in Theorem 6.6 the construction of a compact set Kε = K((Mp)p) withνφc∗fc (E) < ε and ν(E) < ε for E ⊂ Σ − Kε. On the other hand, for E ⊂ Kε wehave the following.

Claim 8.3. There is a finite set S ⊂W1 and a sequence w(k) ∈ Ws,a, s ∈ S with

νφc∗fc (E) =

∞∑k=1

νφc∗fc ([w(k)])

ν(E) =

∞∑k=1

ν([w(k)])

Proof of claim. Inductively define u(B,n) to be the shortest word with [Bu(B,n)A] ⊂E − ∪k<n[Bu(B, k)A]. Either the sequence terminates or |u(B,n)| → ∞ by com-pactness of Kε. Then, up to measure zero sets, E =

⋃n[Bu(B,n)A]. �

32 RHIANNON DOUGALL

Let C be the maximum of the constant in Lemma 8.1 and in 12.15. For a cylinder[BwA] the hypothesis that γ = 1 gives

νφc∗fc ([BwA]) ≤ CR|w|(Bwx)

≤ C2ν([Bwa]).

In conclusion, we choose η = min(ε, C2ε). �

9. Proof of Theorems 3.13 and 3.14

Lemma 9.1. Assume that f is some vector with γ(φ ∗ f) = 1. Let νφc∗fc be given.Then, for any bounded H ∈ L1(µ) and any B ∈ W1,∫

[B]

1

N

N∑n=1

H ◦ σn(z)dνφc∗fc (z)→ µ(H)νφc∗fc ([B]) as N →∞.

Proof. Let H ∈ L1(µ), bounded, and B ∈ W1. As µ is ergodic we have

1

N

N−1∑n=0

H ◦ σn(z)→ µ(H) as N →∞;

pointwise convergence for µ almost every z. In particular

1B(z)1

N

N−1∑n=0

H ◦ σn(z)→ µ(H)1B(z) as N →∞;

pointwise convergence for µ almost every z. And we have the domination

1B(z)1

N

N−1∑n=0

H ◦ σn(z) ≤ ‖H‖∞1B(z),

As νφc∗fc is absolutely continuous with respect to µ we have that 1BH ◦ σn ∈L1(νφc∗fc ) and pointwise convergence of Birkhoff sums νφc∗fc -almost surely. Wecheck integrability of the domination∫

1B(z)1

N

N−1∑n=0

H ◦ σn(z)dνφc∗fc (z) ≤ ‖H‖∞νφc∗fc ([B]) <∞.

In conclusion, using the Dominated Convergence Theorem,∫1B(z)

1

N

N−1∑n=0

H ◦ σn(z)dνφc∗fc (z)→ µ(H)

∫1B(z)dνφc∗fc as N →∞.

�

Let us give a different description of σr∗νφc∗fc . Suppose awA is admissible. We

have1[awA] ◦ σr =

∑B∈W1

∑v∈WB,a

r

1[vwA].

. By Lemma 8.1 we have∫1[awA](σ

rz)dνφc∗fc (z)

≤ lim supk→∞

∑B∈W1

∑v∈WB,a

r

CRr+|w|−1(vwx)〈ρ(s(w))φ[tk] ∗ f, ρ(s(v))−1φ[tk] ∗ f〉

‖φ[tk] ∗ f‖2

Lemma 9.2. Assume that γ(φ ∗ f) = 1. Let Q be a finite subset of Wa,A, and foreach awA ∈ Q let q(w) ∈ (0,∞). There is a constant C > 0 with∫ ∑

awA∈Q

q(w)

R|aw|(awx)1[awA]◦σr(z)1[B](z)dν

φc∗fc (z) ≤ C‖

∑awA∈Q

q(w)1[awA]ρ(s(w))‖.


Proof.∫ ∑awA∈Q

q(w)

R|aw|(awx)1[w] ◦ σr(z)1[B](z)dν

φc∗fc (z)

= limk→∞

t|w|k

∫ ∑awA∈Q

q(w)

R|aw|(awx)1[w] ◦ σr(z)1[B](z)dν

φc∗fc [t](z)

≤ lim supk→∞

C∑

awA∈Q

q(w)

R|aw|(awx)

∑v∈WB,a

r

t|w|k t−r−|w|+1k Rr+|w|−1(vwx)

〈ρ(s(w))φ[tk] ∗ f, ρ(s(v))−1φ[tk] ∗ f〉‖φ[tk] ∗ f‖2

≤ lim supk→∞

C2∑

v∈WB,ar

t−r+1k Rr−1(vx)

〈∑awA∈Q q(w)ρ(s(w))φ[tk] ∗ f, ρ(s(v))−1φ[tk] ∗ f〉

‖φ[tk] ∗ f‖2

≤ lim supk→∞

C2∑

v∈WB,ar

t−r+1k Rr−1(vx)‖

∑awA∈Q

q(w)ρ(s(w))‖

Now we use the fact that tk > γ(φ ∗ f) = 1 to conclude

supr∈N

lim supk→∞

∑v∈WB,a

r

t−r+1k Rr−1(vx) <∞.

�

Proposition 9.3. There is v ∈ H+ with

sup06=f∈H+

γ(φ∗ ∗ f) = γ(φ∗ ∗ v).

Proof. Let fk be unit vectors approaching the supremum. Set v =∑∞k=1 2−kfk.

By positivity∫〈ρ(g)v, v〉Hd(φ∗ ∗ φ)[t] ≥ 2−2k

∫〈ρ(g)fk, fk〉Hd(φ∗ ∗ φ)[t]. In par-

ticular, for any t > γ(φ ∗ρ v) we have∫〈ρ(g)v, v〉Hd(φ∗ ∗ φ)[t] converges and so∫

〈ρ(g)fk, fk〉Hd(φ∗ ∗φ)[t] converges giving t ≥ γ(φ∗ρ fk). In conclusion γ(φ∗ρ v) ≥γ(φ ∗ρ fk) for every k. �

Proof of Theorem 3.13. By Proposition 9.3 it suffices to prove that the supremumof γ(φ ∗ v) is strictly less that 1. We proceed by contradiction, assume that f issome vector with γ(φ ∗ f) = 1. Using transitivity we may choose Q and q(w) with∑awA∈Q q(w)ρ(s(w)) = Mk

p , where Mp is a symmetric random walk operator on

G. In particular there is some β < 1 with ‖Mp‖ < βk.

Set H =∑awA∈Q

q(w)R|w|(wx)1[awA]. Note that H is bounded because Q is finite.

By Lemma 9.1,∫1

N

N−1∑n=0

H ◦ σn(z)1[B](z)dνφ∗fc (z)→ µ(H)νφ∗fc ([B])

as N →∞. We compute that

µ(H) =∑

awA∈Q

q(w)

R|w|(wx)

∫1[awA]hdν

≥∑

awA∈Q

q(w)

R|w|(wx)(h(z)− βw∧z)ν([w])

≥ (h(z)− βw∧z)C−1.

This is a contradiction to the upper bound Cβk. �

Proof of Theorems 3.14. Aiming for a contradiction, we assume that supH γ(φ ∗1eH ) = 1. Let ρ be the direct sum representation. Then there is f in the direct

34 RHIANNON DOUGALL

sum with γ(φ ∗ f) = 1. Using the visibility hypothesis we may choose Q and q(w)so that ∑

awA∈Qq(w)ρ(s(w))v ≤

∑u1,u2∈F

ρ(u1)Mkp ρ(u2)

for a finite set F . Now the upper bound is∑awA∈Q

q(w)

R|w|(wx)

1

N

N−1∑n=0

∫1[awA] ◦ σn(z)dνφc∗fc (z)

≤ C2(#F )2βk

Again, we have arrived at a contradiction. �

10. A shift invariant φ∗ (twisted) measure

In order to construct the shift invariant measure we prefer to work in the twosided shift space Σ, and we will need to assume that the phase space is compact.Therefore, assume throughout that Σ is a subshift of finite type. We also alwaysassume that Ts is transitive.

We set up some notation. There is a basis of open sets given by cylinders

[u1 · · ·um.v1 · · · vn] = {z ∈ Σ : zi = vi+1, 0 ≤ i ≤ n− 1, z−i = um+1−i, 1 ≤ i ≤ m}For any subset Λ ⊂ Z we will say that a sequence y : Λ → W1 is admissible ifτ(yi, yi+1) = 1 for all i ∈ Λ. For Y : Z<0 → W1 and u = u1 · · ·um a finite wordwe denote the concatenation Y u : Z<0 → W1, (Y u)i = um+1−i if 1 ≤ i ≤ m and(Y u)i = Ym+1−i otherwise. Similarly, for y : Z≥0 → W1 and v = v1 · · · vn a finiteword we denote the concatenation vy : Z≥0 →W1, (vy)i = vi+1 if 0 ≤ i ≤ n−1 and(vy)i = yi−n otherwise. The concatenation of Y : Z<0 →W1 and y : Z≥0 →W1 isdenoted Y.y : Z→W1, (Y.y)i = Yi if i < 0 and (Y.y)i = yi if i ≥ 0. We assume thatR : Σ→ R depends only on future coordinates, meaning for any z = Y.y z′ = Y ′.y,we have R(z) = R(z′). We denote the common value as R(.y). The Dirac mass atan arbitrary z ∈ Σ is denoted D(z).

The normalizing factor is

F (φ∗)[t] = limN→∞

〈φc;≤N [t] ∗ φc;≤N [t] ∗ f, f〉.

Definition 10.1. For t > γ(φ, f) and N ∈ N denote,

µφ∗∗fc;N [t] =

∑B,b∈W1:τ(B,b)=1

N∑n=1

t−ncn

N∑m=1

t−mcm

∑v∈Wb,a

n

Rn(vx)∑

u∈WA,Bm

Rm(uvx)〈ρ(s(u)s(v))f, f〉

F (φ∗)[t]D(Xu.vx)

for fixed x : Z≥0 →W1, X : Z<0 →W1 chosen with xi = A uniquely at i = 0, andXi = a uniquely at i = −1.

The measure µφ∗∗fc;N [t] is a positive finite linear combination of Dirac masses and

is therefore well defined. By definition, the mass of µφ∗∗fc;N [t] is bounded for each

t and the mass of [a.A] tends to 1 as N → ∞. By weak* compactness there areaccumulation points. We check convergence along N and a rearrangement for thelimiting measure.

Lemma 10.2. There is weak convergence of µφ∗∗fc;N [t] as N → ∞. In addition the

weak limit coincides with µφ∗∗fc [t] defined by

µφ∗∗fc [t] =

∞∑N=1

t−N∑

w∈WA,aN

RN (.wx)〈ρ(s(w))f, f〉F (φ∗)[t]

N−1∑k=1

ckcN−kσk∗D(X.wx).


Proof. First, let us assert that µφ∗∗fc;N [t] weak* converge as N → ∞. Indeed for t

fixed, any cylinder E has that µφ∗∗fc;N [t](E) is a monotonically increasing bounded

sequence in N .The limit of the measure of any cylinder E is

1

F (φ∗)[t]limM→∞

M∑N=1

t−N∑

m,n≤N∑B,b∈W1:τ(B,b)=1

∑v∈WB,a

n

∑u∈WA,b

m

Rm+n(uvx)〈ρ(s(uv))f, f〉F (φ∗)[t]

cncmσn∗D(X.uvx)(E)

It is clear there is an lower (resp. upper) bound by µM/2(E) (resp. µM (E)) givenby

µM (E) =1

F (φ∗)[t]limM→∞

M∑N=1

t−N∑

w∈WA,aN

RN (wx)〈ρ(s(w))f, f〉N−1∑k=1

ckcN−kσk∗D(X.wx)(E).

Hence for any cylinder E,

µφ∗

c;M [t](E) ≥ µM/2(E) ≥ µφ∗

c;M/2[t](E).

Using convergence along M and M/2 the conclusion follows. �

Definition 10.3. Let c be slowly increasing with

limN→∞

〈φc;≤N [t]∗φc;≤N [t]∗f, f〉 → ∞ and limN→∞

〈φc;≤N [t] ∗ φc;≤N [t] ∗ f, f〉〈φc[t] ∗ f, f〉

→ ∞ as t→ γ(f).

Any accumulation point of µφ∗∗fc [tk] as tk → γ(f) is denoted µφ

∗

c and called a φ∗

(twisted) measure.

10.1. Shift invariance.

Theorem 10.4. Any φ∗ (twisted) measure µφ∗

c is σ-invariant.

The proof of shift invariance will rely on a technical lemma regarding the slowlyincreasing function. For z ∈ Σ set

P(z, c ∗ c,N) =

N−1∑k=1

ckcN−kσk∗D(z).

Then

µφ∗∗fc [t]−σ∗µφ

∗∗fc [t] =

∞∑N=1

t−N∑

w∈WNa,a


(P(z, c ∗ c,N)− σ∗P(z, c ∗ c,N))

Claim 10.5. There is γk → 0 with

|P(z, c ∗ c,N)(E)− σ∗P(z, c ∗ c,N)(E)| ≤ γkP(z, c ∗ c,N)(E) + 6k(cNck + cNck)

Proof. Write τ−1cm = cm−1 τ+1cn = cn+1.

σ∗P(z, c ∗ c,N) =

N∑k=2

ck−1cN−(k−1)σk∗D(z)

=

N−1∑k=1

τ−1ckτ+1cN−kσk∗D(z)− τ−1c1τ+1cN−1σ∗D(z) + τ−1cNτ+1c0σ

N∗ D(z)

We aim to show that∑N−1k=1 τ−1ckτ+1cN−kσ

k∗D(z) = P(z, τ−1c ∗ τ+1c,N) is close to

P(z, c ∗ c,N).Since c is increasing we have τ−1ck ≤ ck and ck ≤ τ+1ck, giving

[c ∗ c− τ−1c ∗ τ+1c]N ≤ [(c− τ−1c) ∗ c]N , [τ−1c ∗ τ+1c− c ∗ c]N ≤ [c ∗ (τ+1c− c)]N .

36 RHIANNON DOUGALL

Choose K sufficiently large so that cn − cn−1 ≤ εcn−1 and cn − cn−1 ≤ εcn−1 forn > K. Then

[(c− τ−1c) ∗ c]N ≤ εc ∗ cN −K∑k=1

(ck − τ−1ck)cN−k ≤ εc ∗ cN + 2KcKcN

and

[c ∗ (τ+1c− c)]N ≤ εc ∗ cN −N∑

k=N−K

ck(τ+1cN−k− cN−k) ≤ εc ∗ cN + 2(1 + ε)KcNcK

Putting these together gives

|P(z, c ∗ c,N)(E)− σ∗P(z, c ∗ c,N)(E)| ≤ |P(z, c ∗ c,N)(E)− P(z, τ−1c ∗ τ+1c,N)(E)|+ c1cN

≤ εP(z, c ∗ c,N)(E) + 2(1 + ε)KcKcN + c1cN

�

Proof of Theorem 10.4. Let E be an open set. We will show that for for every ε > 0we have |µφ∗∗fc [t]− σ∗µφ

∗∗fc [t](E)| ≤ ε for t sufficiently close to γ(f). Assuming the

claim, choose k with 2γk maxt≥γ(f) µφ∗∗fc [t](E) ≤ ε. We have

|µφ∗∗fc [t]− σ∗µφ

∗∗fc [t](E)| ≤ γkµφ

∗∗fc [t](E) +

∞∑N=1

t−N∑

w∈WNa,a


6kcNck

= γkµφ∗∗fc [t](E) + 6k

〈φc[t] ∗ f, f〉F (φ∗)[t]

.

By Lemma 6.4, when k is fixed, for t sufficiently close to γ(f) we have 6k 〈φc[t]∗f,f〉F (φ∗)[t] ≤ε2 , whence

|µφ∗∗fc [t]− σ∗µφ

∗∗fc [t](E)| ≤ ε.

�

10.2. The (limit of) matrix coefficients. In the goal of checking conformalproperties it is easier to work with a measure that is one-sided. Recall that R isLipschitz in the θ-metric (see equation 12.1). It follows that there is a constantC > 1 so that for any u and any word b = b1 · · · bp with ub admissible we have

(10.1) C−θp

≤R|u|(uy)

R|u|(uz)≤ Cθ

p

for all y, z ∈ [b1 · · · bp].For brevity write m = π∗µ

φ∗∗fc and Lip(p) = Cθ

p

.

Lemma 10.6. The measures σ(a)∗ m and ν

φ∗c∗fc are equivalent. For any w with

|w| = n and awA admissible we have,

Lip(p)−1 ≤

∫[wA]

R−1n dν

φ∗c∗fc∫

[awA]R−1n ◦ σdm

≤ Lip(p)

for p = w ∧ x.

Proof. Let awA be given with |w| = k. Write p = w ∧ x. Both statements areverified upon checking the second. Suppose w = w′B. From the definition of m we


have

m(R−1k 1[awA]) = µφ

∗∗fc (R−1

k ◦ σk1[aw.A])

= limN→∞

∑B,b∈W1:τ(B,b)=1

N∑n=1

t−ncn

N∑m=1

t−mcm

∑v∈Wb,a

n

Rn(vx)∑

u1···um∈WA,Bm

Rm(uvx)

Rk(um−k · · ·umvx)

〈ρ(s(u)s(v))f, f〉F (φ∗)[t]

1[aw.A](Xu.vx)

Let g = s(w). Using Holder continuity we have an upper bound (lower bound) bya Lip(p)−1 (Lip(p)) multiple of the lim sup (lim inf) of

N∑m=k

t−mcm∑

u∈WA,am

Rm−k(uax)〈ρ(g)φc;≤N ∗ f, ρ(s(u)−1f〉

F (φ∗)[t]

+ rem.Q∗

c,N ;[1,k](R−1k )

where rem.Q∗

c,N ;I is defined for the anti-homomorphism ρ∗; and the first term is equal

to νgQ∗

N ;d ([A]) + rem.gQ∗

c,N ;[N−k,N ]([A]) defined for the anti-homomorphism ρ∗ (see by

Remark 7.7). It follows that

Lip(p)−1 limq→∞

limN→∞

〈ρ(g)φc;≤N ∗ f, φ∗c;≤Nf〉F (φ∗)[t]

≤ m(R−1k 1[awA])

≤ Lip(p) limq→∞

limN→∞

〈ρ(g)φc;≤N ∗ f, φ∗c;≤Nf〉F (φ∗)[t]

Now by Lemma 7.10 and since g = s(w),

limq→∞

limN→∞

〈ρ(s(w))φc;≤N ∗ f, φ∗c;≤Nf〉F (φ∗)[t]

= νφ∗c∗fφc

(R−1k 1[wA]).

�

Remark 10.7. If R is depends on one letter and Σ is the full shift then in fact

σ(a)∗ m = ν

φ∗c∗fc .

Corollary 10.8.

Lip(p)−2 limq→∞

limN→∞

〈ρ(s(w))φc;≤N [tq] ∗ f, φ∗c;≤N [tq] ∗ f〉〈φc;≤N [tq] ∗ f, φ∗c;≤N [tq] ∗ f〉

≤ µφ∗∗fc ([a.wA])

Rk(wx)

≤ Lip(p)2 limq→∞

limN→∞

〈ρ(s(w))φc;≤N [tq] ∗ f, φ∗c;≤N [tq] ∗ f〉〈φc;≤N [tq] ∗ f, φ∗c;≤N [tq] ∗ f〉

11. Amenability implies φ ∗ φ (twisted) measure is Gibbs

11.1. The periodic point variant. In the goal of finding a shift invariant measureit would be more obvious to define a measure supported on periodic points. In orderto use equidistribution arguments from the thermodynamic formalism we will findourselves preferring combinations of periodic points. Define

µφ∗∗fc;N [t] =

N∑n=1

t−n∑

n=m+k

cmck∑

z∈Σ:σnz=z

Rn(z)〈ρ(s(z0 · · · zn))f, f〉

F (φ∗)[t]D(z).

38 RHIANNON DOUGALL

Denote dn =∑m+k=n cmck and

φd;≤N [t](g) =

N∑n=1

t−n∑

n=m+k

cmck∑

z∈Σ:σnz=z,s(z1···zn)=g

Rn(z)s(z0 · · · zn)).

It is clear that any accumulation point of µφ∗∗fc;N [t] has mass limN→∞

〈φd;≤N [t]∗f,f〉F (φ∗)[t] .

We check absolute continuity and boundedness of the mass of the measure. We usethe notation

F �C H ⇐⇒ C−1F ≤ H ≤ CF,

for a constant C > 1.

Lemma 11.1. There is a constant C so that for any cylinder [WB.bw] we have

µφ∗∗fc ([WB.bw]) �C µφ

∗∗fc ([WB.bw]).

Proof. Write µφ∗∗fc = µA,a,x (with A, a, x defining φ). To begin with we denote

E = [WB.bw]. For each D ∈ W1 choose a one-sided xD ∈ [D] ∩ Σ+. Using Holdercontinuity we have,

µφ∗∗fc;N [t](E)

=

N∑n=1

t−n∑

n=m+k

cmck∑

z:σnz=z

Rn(z)〈ρ(s(z0 · · · zn))f, f〉

F (φ∗)[t]1E(σmz)

=∑

D′,D:τ(D′,D)=1

N∑n=1

t−n∑

n=m+k

cmck∑

z:σnz=z,zn=D′,z0=D

Rn(z)〈ρ(s(z0 · · · zn))f, f〉

F (φ∗)[t]1E(σmz)

�CN∑n=1

t−n∑

n=m+k

cmck∑

D′,D:τ(D′,D)=1

∑u∈WD′,B

m

∑v∈Wb,D

k

Rm+k(.uvxD)〈ρ(s(uv))f, f〉F (φ∗)[t]

1E((vu)∞u.v(uv)∞)

where (uv)∞ (resp. (vu)∞) is the right-infinite (resp. left-infinite) concatenationof uv (resp. vu). Now recall that E = [WB.bw], and suppose that |WB| = P ,|bw| = p. Then

∑D′,D:τ(D′,D)=1

µD′,D,xD

c;N [t](E)−〈φb,ac;≤p[t] ∗ f, (φ

A,Bc;≤N )∗[t] ∗ f〉

F (φ∗)[t]−〈φb,ac;≤N [t] ∗ f, (φA,Bc;≤P )∗[t] ∗ f〉

F (φ∗)[t]

≤2N∑n=1

t−n∑

n=m+k

cmck∑

D′,D:τ(D′,D)=1

∑u∈WD′,B

m

∑v∈Wb,D

n

Rm+k(.uBUV bvxD)〈ρ(s(V Bwbv))f, f〉

F (φ∗)[t]

≤∑

D′,D:τ(D′,D)=1

µD′,D,xD

c;2N [t](E)

The limit in N and 2N agrees. Taking a limit in t makes the remainder term vanish.Then it is straightforward to check that µD,d,xd is equivalent to µA,a,x. �

For the remainder of this section we have only results for the case f = δe. Recallthat φc ∗ δe = φc and in particular 〈φd[t] ∗ δe, δe〉 = φd[t](e).

For amenable groups the work of [9] shows that the Gurevic pressure P (logR, Ts)is equal P (logR + ψ) for a unique real one-dimensional representation π : G → Rand ψ = π ◦ s. (In other words: let ψab be the composition of s with G → G/[G :G] ∼= Zd ⊕ F → Zd, with F the finite torsion group. Then there is a unique ξ ∈ Rthat determines ψ(x) = 〈ξ, ψab(x)〉Rd .) We use similar ideas to [16] describing driftfor abelian extensions; and the same ideas behind the equidistribution result of [9].


Our goal is to prove Theorem 3.16: if G is amenable then any φ∗ (twisted) measureµφ∗

c is equal to the equilibrium state µR expψ; morever for each g

limk→∞

limN→∞〈ρ(g)φc;≤N [tk], φ∗c;≤N [tk]〉〈φc;≤N [tk], φ∗c;≤N [tk]〉

= ψ(g).

We use the following large deviation estimate

Lemma 11.2. Let K be a weak∗ compact set not containing µR expψ. Then

1

nlog

∑{(w)=x∈Per(n),s(w)=e, 1n

∑n−1k=0 Dσkx∈K}

Rn(x)→ β < P (logR+ ψ)

Proof. We allow ourselves to use ν to denote an arbitrary shift invariant measure(previously it was reserved for the R conformal measure). Set

ρ = infν∈K

supF

(∫Fdν − P (logR+ ψ + F )

)We claim that ρ+ P (logR+ ψ) > 0. First, for any F(∫

Fdν − P (F + logR+ ψ) + P (logR+ ψ)

)= −

(P (F + logR+ ψ)−

∫F + ψ + logRdν

)+ P (logR+ ψ)−

∫ψ + logRdν

and so

supF

(∫Fdν − P (F + logR+ ψ) + P (logR+ ψ)

)= − inf

F

(P (F + logR+ ψ)−

∫F + ψ + logRdν

)+

(P (logR+ ψ)−

∫ψ + logRdν

)= −h(ν) +

(P (logR+ ψ)−

∫ψ + logRdν

)> 0

Since infF(P (F + logR+ ψ)−

∫F + ψ + logRdν

)= infG

(P (G)−

∫Gdν

)= h(ν)

and is strictly positive by uniqueness of the equilibrium state. The lower bound isuniform in ν in K by lower semi-continuity.

Now, by definition of ρ, for every ν ∈ K we have

supF

∫Fdν − P (F + logR+ ψ) > ρ

and so we may choose γ > 0 and F with∫Fdν − P (F + logR+ ψ) > ρ− γ

We deduce that

K ⊂{ν :

∫Fdν − P (F + logR+ ψ) > ρ− γ

},

and since K is weak∗ compact there are F1, . . . , Fk with

K ⊂k⋃i=1

{ν :

∫Fidν − P (Fi + logR+ ψ) > ρ− γ

}.

We need the following two observations

(11.1) τx,n =

n−1∑k=0

Dσkx ∈ K =⇒ exp(Fni (x)−nP (Fi + logR+ψ)−nρ+nγ) ≥ 1,

(11.2) (w) = x ∈ Per(n), s(w) = e =⇒ ψn(x) = 0.

40 RHIANNON DOUGALL

Putting this together gives∑{(w)=x∈Per(n),s(w)=e, 1n


Rn(x)

≤∑

{(w)=x∈Per(n),s(w)=e, 1n∑n−1k=0 Dσkx∈K}

Rn(x) exp(−nP (Fi + logR+ ψ)− n(ρ− γ) + Fni )

≤ exp(−nP (Fi + logR+ ψ)− n(ρ− γ))∑

{(w)=x∈Per(n),s(w)=e, 1n∑n−1k=0 Dσkx∈K}

Rn(x) exp(Fni + ψn)

So

1

nlog

∑{(w)=x∈Per(n),s(w)=e, 1n


Rn(x)

≤ −P (Fi + logR+ ψ)− ρ+ γ + P (logR+ ψ + Fi)

≤ −ρ+ γ

< −ε+ P (logR+ ψ) + γ.

�

Proof of Theorem 3.16. Let H : Σ → R be continuous and non-negative. Let K ={m :

∣∣∫ Hdm− ∫ HdµR expψ

∣∣ ≥ ε}, a compact set that clearly does not contain

µR expψ. Writing τx,n = 1n

∑n−1k=0 Dσkx,

tn∑

x∈Per(n),ψn(x)=0,τx∈K

Rn(x)

∫Hdτx,n ≤ ‖H‖∞tn

∑x∈Per(n),ψn(x)=0,τx,n∈K

Rn(x),

∞∑n=1

dntn

∑x∈Per(n),ψn(x)=0,τx∈K

Rn(x)

∫Hdτx,n ≤ ‖H‖∞

∞∑n=1

dntn

∑x∈Per(n),ψn(x)=0,τx,n∈K

Rn(x),

By Lemma 11.2 the series on the right converges at t = P (logR + ψ), denote thevalue as C.

Therefore,

µφ∗

c [t](H) =1

F (φ∗)[t]

∞∑n=1

dntn

∑(w)=x∈Per(n)

〈ρ(s(w))δe, δe〉Rn(x)

∫Hdτx,n

≤ ‖H‖∞F (φ∗)[t]

∞∑n=1

dntn

∑(w)=x∈Per(n),s(w)=e,τx,n∈K

Rn(x)

+1

F (φ∗)[t]

∞∑n=1

dntn

∑(w)=x∈Per(n),s(w)=e,τx,n /∈K

Rn(x)

(∫HdµR expψ + ε

)

≤ ‖H‖∞C

F (φ∗)[t]+

(∫HdµR expψ + ε

)limN→∞ φd;≤N [t](e)

F (φ∗)[t]

A lower bound follows similarly. We conclude

limk→∞

limN→∞ φd;≤N [tk](e)

limN→∞ φc;≤N [tk] ∗ φc;≤N [tk](e)µφ∗φδe

(H) =

∫HdµR expψ.

Up to scaling, the measures coincide. (Finiteness of the scaling is given by Lemma11.1.) Now it follows that µφ

∗

c , µφ∗

c and µR expψ are equivalent measures, and byergodicity are proportional.

We are ready to harvest the (limit of) matrix coefficients property. For every pwe can choose u, v with |u| = |v| = n ≥ p, u ∧ v ≥ p, u ∧ x ≥ p and s(u) = g,


s(v) = e. Now we have, on the one hand using 12.18,

µR expψ(R−1n 1[u])

µR expψ(R−1n 1[v])

∈ exp(ψ(g))[α−1p , αp].

And on the other hand using Corollary 10.8

µφ∗

c (R−1n 1[u])

µφ∗φδe(R−1

n 1[v])∈

limN→∞〈ρ(g)φc;≤N [tk], φ∗c;≤N [tk]〉limN→∞〈φc;≤N [tk], φ∗c;≤N [tk]〉

[(1+ε(tk))Lip(p)−2, (1−ε(tk))Lip(p)2],

where ε(tk)→ 0. Since p was arbitrary this completes the proof. �

12. Countable Markov shifts and strong positive recurrence

In this section we expand on the basic machinery for countable Markov shifts.Our use of the notion of strong positive recurrence is non-standard, and moreoverour choice of “first-return series” is non-standard. It will be important to makeclear the allowed estimations regarding local Holder continuity. Equilibrium statesin a CMS need not satisfy the Gibbs property but it is still possible to estimateratios of certain cylinders, as we make clear.

12.1. Basic definitions. Let x ∈ NZ; that is, x is a bi-infinite sequence in thecountable set N — we will often write W1 in place of N, and describe W1 as thealphabet of the CMS. As is common usage, we use xi to denote the ith element inthe sequence. In the theory of Markov shifts, it is usual to write x = . . . x−1.x0x1 . . .;that is, to separate the negative coordinates from the non-negative coordinates by aperiod. In order to define a CMS it is useful to make reference to a transition matrixτ : W1 × W1 → {0, 1}. The (two-sided) countable Markov shift (with transitionmatrix τ) is

Σ ={x ∈ WZ

1 : τ(xi, xi+1) = 1}.

We use σ to denote the left shift σ : Σ→ Σ, (σx)i = xi+1. We always assume thatthe dynamics are transitive for σ : Σ→ Σ. The (two-sided) countable Markov shift(with transition matrix τ) is

Σ+ ={x ∈ WN

1 : τ(xi, xi+1) = 1∀i ∈ N}.

We can project from Σ to Σ+ by “forgetful” map. Many of the constructions thatfollow naturally pass from Σ to Σ+.

We equip Σ with the product topology, which can be metrized: write x ∧ y =min(k + 1 : xi = yi,−k ≤ i ≤ k), and d(x, y) = 2−x∧y. We use some convenientnotation for a basis of open balls,

[W1 · · ·Wm.w1 · · ·wn] = {x ∈ Σ : xj = wj , j = 0, . . . , n, x−m−1+i = Wi, i = 1, . . . ,m}

where wj , wj+1,Wi,Wi+1 ∈ W1, τ(wj , wj+1) = 1, τ(Wi,Wi+1) = 1, τ(Wm, w0) = 1,i = 0, . . . , n−1, j = 1, . . . ,m−1. We say that w = w1 · · ·wn, W = W1 · · ·Wn, Wware admissible, and |w| = n, |W | = m. For B, b ∈ W1 we write

WB,bn = {Bu1 · · ·un−2b : uj , uj+1 ∈ W1, τ(uj , uj+1) = 1, j = 1, . . . , n− 3}

and

WB,b = {Bu1 · · ·unb : uj , uj+1 ∈ W1, τ(uj , uj+1) = 1, j = 1, . . . , n;n ∈ N}

Let R+ : Σ+ → R+ be a positive function. We say that logR+ is locally Holdercontinuous if there is θ < 1 for which

Varn(logR+) = sup{| logR+(x)− logR+(y)| : x ∧ y = n

}has

(12.1) Varn(logR+) ≤ Cθn

42 RHIANNON DOUGALL

for n > 1. It should be noted that logR+ can be unbounded on any cylinder whenΣ+ is non-compact. We lift R+ to Σ by defining R(· · ·x−1.x1 · · · ) = R+(x1 · · · ).We will simply write R for both functions.

We write const.(Bb) for the constant with

(12.2) (const.(Bb))−1 ≤ Rk+1(uBby)

Rk+1(uBbz)≤ const.(Bb)

for any admissible uBb, |u| = k, |b| = |B| = 1, and y, z ∈ σ[b]. Local Holdercontinuity implies that const.(Bb) does not depend on B, b. Nevertheless we usethis notation to remind us to condition on two letters, as local Holder demands.

12.2. Strong positive recurrence by discriminants. Following [20] we say thatR is positive recurrent if there is a constant Ma with∑

x:σnx=x,x0=B

Rn(x) ∈ [M−1B expnP (logR),MB expnP (logR)]

for all n ∈ N.The definition of strong positive recurrence is in introduced by Sarig [21] in

terms of discriminants. Let us borrow some of the notation for this discussion. Weswitch to additive notation (e.g. R = exp r for r locally Holder continuous). Writeϕa = 1[a](x) inf {n ≥ 1 : σnx ∈ [a]},

(12.3) Zn(r) =∑σnx=x

(exp r)n(x)1[a](x) ; Z∗n(r) =∑σnx=x

(exp r)n(x)1[ϕa=n](x)

(12.4) P (r) := P (r, σ) = lim supn→∞

1

nlogZn(r) ; −p∗ := lim sup

n→∞

1

nlogZ∗n(r)

We write P (r) for the pressure in the system induced on returning to a. (So

r = exp(∑ϕa−1k=0 r ◦ σn ◦ π), and π maps the induced phase space to [a] ⊂ Σ+.)

The Discriminant Theorem (Theorem 2 of [21]) states that P (r + p) = 0 has aunique solution p = −P (r) if r is not transient (“not transient” is the same as “re-current”). The first return series controls the range of p for which P (r + p) <∞, indeed sup {p : P (r + p) <∞} = p∗. The a-discriminant of r is ∆a[r] =

sup {P (r + p) : p < p∗}. Non-transience implies that ∆a[r] ≥ 0, and indeed P (r − P (r)) =0. Following Sarig [21], one says that r is strongly positively recurrent if ∆a[r] > 0.(At ∆a[r] = 0 the behaviour can be either positive recurrence or null recurrence.)

The main thing for us to note is Proposition 3 that P (r + p) is strictly increasingin (−∞, p∗]. Now if P (r) 6= −p∗ then there is some −P (r) < t < p∗ with 0 <P (r + t) <∞ from which ∆a[r] > 0. Conversely, if P (r) = −p∗ then ∆a[r] = 0. Inconclusion, strong positive recurrence is equivalent to

(12.5) P (r) 6= −p∗,

and this is the formulation we will use.We will use the notation

(12.6) − p∗ = γ(SPR).

12.3. Return series.

Definition 12.1. We will say that η is a power series in t−1 if η is a function of areal variable t ∈ [0,∞) defined as

η[t] = limN→∞

N∑n=0

t−nan

with an ∈ [0,∞).


Observe that η[t] coincides with the Dirichlet series∑∞n=1 exp(−sn)an evaluated

at s = log t and so we borrow terminology such as abscissa of convergence butapplied to the variable t = exp(s). As η[t− ε] ≥ η[t] for any 0 < t− ε < t we havethat

{t > 0 : η[t] <∞} ⊂ [σ,∞),

(0, σ) ⊂ {t > 0 : η[t] =∞} ,

for some σ ∈ [0,∞] which we call the abscissa of convergence (of the power seriesin t−1).

Let R : Σ+ → R+ with Σ+ some CMS and logR locally Holder continuous.Recall we always assume that R has P (logR, σ) = 0. To begin with we do not evenask that R is recurrent — in this way the discussion applies to both σ : Σ→ Σ andto the skew product Ts : Σ × G → Σ × G, upon realising an isomorphism with aCMS. For a letter B ∈ W1, the (B-conditioned) (periodic) return series is

(12.7) t 7→∞∑n=1

t−n∑

x:σnx=x,x0=B

Rn(x),

where Rn(z) = R(z)R(σz) · · ·R(σn−1z). Using local Holder continuity 12.7 has thesame abscissa of convergence as the (B-conditioned) return series

(12.8) t 7→∞∑n=1

t−n∑

b:τ(b,B)=1

∑w∈WB,b

n

Rn(wBy),

with fixed initial condition y ∈ σ[B]. And indeed if σ is transitive then the abscissaof convergence is equal to expP (logR, σ) = 1.

The periodic B first return series,

(12.9) t 7→∞∑n=1

t−n∑

x:σnx=x,x0=B,xi 6=B,0<i<n

Rn(x),

is within a constant multiple of

(12.10)

∞∑n=1

t−n∑

b:τ(b,B)=1

∑w∈WB,b

n :wi 6=B,i>1

Rn(wBy).

We say that R has a growth gap if 12.9, or equivalently 12.10, converges for t = 1−ε,for some ε > 0. The existence of a growth gap will allow us to consider two-letterconditioned returns.

Fix A, a ∈ W1 with aA admissible. Define the (A, a conditioned) return seriesas

(12.11) ζA,a[t] =

∞∑n=1

t−n∑

w∈WA,an

Rn(wAx).

with initial condition Ax ∈ Σ+.

Lemma 12.2. If R has a growth gap then{t > 0 : ζA,a[t] <∞

}= 1.

If R is recurrent then ζA,a[1] =∞.

44 RHIANNON DOUGALL

Proof. Notice that∞∑n=1

t−n∑

w∈WA,an

Rn(wAx)

∞∑m=1

t−m∑

b:τ(b,A)=1

∑u∈WA,b

m :ui 6=A,i>1

Rm(uAx)

≥ const.(aA)

∞∑n=1

t−n∑

w∈WA,an

∞∑m=1

t−m∑

b:τ(b,A)=1

∑u∈WA,b

m :ui 6=A,i>1

Rn+m(wuAx)

= const.(aA)

∞∑n=1

∞∑m=1

t−mt−n∑

w∈WA,an

∑b:τ(b,A)=1

∑u∈WA,b

m :ui 6=A,i>1

Rn+m(wuAx)

= const.(aA)

∞∑n=1

∞∑m=1

t−mt−n∑

b:τ(b,A)=1

∑v∈WA,b

n+m

Rn+m(vAx)

≥ const.(aA)

∞∑k=1

t−k∑

b:τ(b,A)=1

∑v∈WA,b

k

Rk(vAx).

�

This says that conditioning on one or two letters contains the same informationin the case that R is has a growth gap. We update the notion of first returns. Wewrite An = {Awa : w ∈ Wn−2, wiwi+1 6= Aa, i = 1, . . . , n− 3}. Define

(12.12) ηA,a[t] :=

∞∑n=1

t−n∑w∈An

eRn(wx).

Then γ(SPR) ≥ inf{t > 0 : ηA,a[t] <∞

}.

We summarize what we have learnt.

Lemma 12.3. Let R : Σ+ → R+ with Σ+ a mixing CMS and logR locallyHolder continuous. Assume R is strongly positively recurrent. Then γ(SPR) <expP (logR) and

(γ(SPR),∞) ⊂{t > 0 : ηA,a[t] <∞

},

(γ(SPR),∞) ⊂{t > 0 : ζA,a≤N [t] <∞

},

(1,∞) ={t > 0 : ζA,a[t] <∞

}.

We are now able to prove Proposition 3.1.

Proof of Proposition 3.1. expP (logR, Ts) > γ(SPR) implies that R : Σ+×G→ R+

(viewed as a CMS) has a growth gap. In particular by Lemma 12.2, for any fixed h

inf{t > 0 : ζ(A,e),(a,h)[t] <∞

}= expP (logR, Ts).

But also φ[t](e) is bounded from above by the (A, e)-conditioned periodic returnseries, and so γ(φ∗ ∗ δe) ≤ expP (logR, Ts). �

12.4. Equilibrium states. Recall that we assume P (logR, σ) = 0. Positive re-currence hypothesis guarantees that the transfer operator L has an eigenfunctionLh = h, with h a positive locally Holder continuous and eigenmeasure L∗ν = ν;see [20]. The equilibrium state dµ = hdν is finite, σ invariant and ergodic.

When Σ is compact µ has the Gibbs property (see [25]): there is a constant C > 0with

(12.13) C−1Rn(wy) ≤ µ([w]) ≤ CRn(wy)

for any w of length n and y ∈ σn[w]. In general we cannot expect to have the Gibbsproperty in the CMS setting but we are able to make use of the conformal propertyfor ν.


Definition 12.4. A measure ν is said to be R-conformal if there is λ with

(12.14)

∫R−1n 1[wb]dν = λ−nν([b])

for any w ∈ Wn and b ∈ W1 with wb admissible.

In particular ν is R-conformal with λ = 1. We also need a local version of aGibbs inequality

Definition 12.5. A measure ν has a RHS local Gibbs inequality if there is a constantCb with

(12.15) ν([wb]) ≤ CbRn(wbx)

for any admissible wb with |w| = n and for any x ∈ σ[b].A measure ν has a local Gibbs inequality if there is a constant Cb with

(12.16) C−1b Rn(wbx) ≤ ν([wb]) ≤ CbRn(wbx)

for any admissible wb with |w| = n and for any x ∈ σ[b].

We check that ν has a local Gibbs inequality: inside the conformal property 12.14we substitute Rn(wbx) ≤ const.(wnb)Rn(wby) using 12.2. This gives

(12.17) const.(wnb)−1Rn(wbx) ≤ ν([wb])

ν([b])≤ const.(wnb)Rn(wbx)

for any x ∈ σ[b].We make a similar estimate for the equilibrium state µ. First recall the standard

manipulations to check shift invariance:∫f ◦ σndµ =

∫f ◦ σnhdν =

∫f ◦ σnhd(L∗)nν

=

∫fLnhdν =

∫fhdν.

Now we check the integral of R−1n 1[w],∫

R−1n 1[w]dµ =

∫R−1n 1[w]hdν

=

∫Ln(R−1

n 1[w]h)dν

=

∫σn[w]

h(w·)dν

≤ h(wξ) + |h|θβn,

where |h|θ is the local Holder constant for h.Suppose R expψ is another locally Holder strongly positively recurrent function

and that ψ depends only on one letter (such is the case for Abelian extensions).Write ν′ the eigenmeasure of R expψ, λ = expP (logR+ψ, σ), h′ the eigenfunction.We have ∫

R−1n 1[w]h

′dν′ = λ−n∫Ln(R−1

n 1[w]h′)dν′

= λ−n∫σn[w]

expψn(w)h′(w·)dν′

≤ expψn(w)λ−n(h′(wξ) + |h′|θβn)

and a lower bound given by expψn(w)λ−n(h′(wξ) − |h′|θβn). Then in particularfor u, v with u ∧ v = p, some fixed ξ ∈ σp([u]), and

αp =(h′(uξ) + |h′|βp)(h′(uξ)− |h′|βp)

,

46 RHIANNON DOUGALL

we have

(12.18) α−1p expψn(u) exp−ψn(v) ≤

∫R−1n 1[u]h

′dν′∫R−1n 1[v]h′dν′

≤ αp expψn(u) exp−ψn(v)

13. Convergence in `2

In this section we present the basic properties relating to convergence of thethermodynamic densities. The main feature is that ρ permutes an orthonormalbasis of the Hilbert space. We use the notation δh to denote the indicator functionon the coset h ∈ G/H.

We begin by showing that for t > γ(SPR), φ≤N [t] belongs to `1(G). Recall thatfor any f : G→ R we define ‖v‖`1(G) =

∑g∈G |v(g)|. (The definition merely uses a

countable series of non-negative terms.) Substituting v = φ≤N [t] in the definitiongives

‖φ≤N [t]‖`1(G) = ζA,a≤N [t],

which we know to be finite by Lemma 12.3.By definition, if t > γ(φ, δe) then supN∈N φ≤N [t](e) < ∞. Using transitivity,

this implies that for each g we have supN∈N φ≤N [t](g) < ∞. A bounded series ofnon-negative terms in R converges and hence φ[t](g) is well-defined (in whicheverway we arrange the series). It is clear that φ[t](g) ≤ ζA,a[t] and so γ(φ, δe) ≤ 1.

We have the identity

‖φ[t]‖`1(G) =∑g∈G

φ[t](g) = ζA,a[t].

So ‖φ[t]‖`1(G) <∞ for t > 1, but ‖φ[t]‖`1(G) = ζA,a[t] =∞ for t ≤ 1.

Proof of Lemma 2.4. Recall that in a Hilbert space we say that XN weakly con-verges to X∞ if 〈XN −X∞, v〉H → 0 as N → ∞ for every vector v. And stronglyconverge if ‖XN − X∞‖H → 0 as N → ∞. The reader may already be familiarwith this fact: if ‖XN‖H is uniformly bounded and XN weakly converge to X∞ onan orthonormal basis then in fact XN strongly converge to X∞. We provide thedetails of this argument. Denote the orthonormal basis vectors permuted by ρ asei for i ∈ N.

The argument is applied to XN = φ≤N [t] ∗ f and X∞ formally defined by〈X∞, ei〉H = limN→∞〈φ[t] ∗ f, ei〉H. Let us check that X∞ is the weak limit withrespect to the orthonormal basis ei. Let v ∈ H+ be arbitrary (it is sufficient tocheck against vectors in H+). We will show that 〈XN , v〉H are a Cauchy sequence.Write v =

∑∞q=1 aqeq. Choose Q with

‖∞∑n=Q

anen‖H ≤ ε/2.

Then we have

〈XN −XM , v〉H ≤∑q≤Q

aq〈XN −XM , eq〉H + 〈XN −XM ,∑q>Q

aqeq〉H.

Using the Cauchy-Schwarz inequality we have

〈XN −XM ,∑q>Q

aqeq〉H ≤ ‖XN −XM‖H‖∑q>Q

aqeq‖H. ≤ ε supn∈N‖Xn‖H

Now for any weak limit Xv∞ (i.e. 〈XN −Xv

∞, v〉H → 0) we also have

ε ≥ 〈XN −Xv∞, v〉H ≥ 〈XN −Xv

∞, aiei〉H,

so that 〈Xv∞, ei〉H = 〈X∞, ei〉H for each i. In conclusion, XN weakly converges to

X∞.


Recall that the norm is lower semicontinuous:

‖X∞‖H = supv〈X∞, v〉H = sup

vlimN〈XN , v〉H ≤ lim inf

n→∞‖XN‖H,

using Cauchy-Schwarz in the last line. It follows that ‖X∞‖H < ∞. On the otherhand we have by the monotonicity

‖X∞‖H ≥ lim supn→∞

‖XN‖H.

Denote Pi>Q the projection on to the span of {ei : i > Q}. We have

‖Pi>Q(X∞ −XN )‖H ≤ ‖Pi>QX∞‖H + ‖Pi>QXN‖H ≤ 2‖Pi>QX∞‖H,

by monotonicity. Using that ‖X∞‖H <∞, we can choose Q large enough to make‖Pi>QX∞‖H < ε/2, for some ε. (Indeed ‖X∞‖H <∞ implies that

∑i∈N |〈X∞, ei〉|2

is a convergent series.) For a fixed Q we can choose N large enough with

‖Pi≤Q(X∞ −XN )‖2H =∑q≤Q

〈X∞ −XN , eq〉2H ≤ ε

In conclusion ‖XN −X∞‖H → 0 as N →∞. That is, we have strong convergenceof XN to X∞. �

Proof of Lemma 2.6. We need to upper bound φB,a in terms of φA,a. Choose uBwith AuBB admissible, in this way any v ∈ WB,a

n has AuBv ∈ WA,an+|uB |+1. More-

over,

R|AuB |+n(AuBvx) = R|AuB |(AuBvx)Rn(vx) ≥ const.(uBB)−1R|AuB |(AuBBz)Rn(vx)

for some fixed z ∈ σ[B]. Let C be the constant with cn ≤ Ccn−|uB |. We deducethat

N∑n=1

t−ncn∑

v∈WB,an

Rn(vx)〈ρ(s(v))f, δh〉

≤ const.(uBB)

R|uB |+1(AuBBz)

N∑n=1

t−ncn∑

v∈WB,an

R|uB |+1+n(AuBvx)〈ρ(s(AuB))−1ρ(s(AuB)s(v))f, δh〉

≤ C const.(uBB)

R|uB |+1(AuBBz)t|uB |+1〈φc;≤N+|uB |[t] ∗ f, ρ(s(AuB))δh〉

The result follows. �

Lemma 13.1. Transitivity implies γ(f) > 0.

Proof. We only need two distinct periodic orbits, represented by u, v with s(u) =s(v) = e, to create non-trivial Gurevic pressure. �

14. The boundary measure for the free group

We explain the action of Fa,b on its visual boundary in terms of a subshift offinite type, the related unitary representation and spherical function.

Identify the visual boundary of Fa,b with Σ+, then an element x is an infinite re-duced word, and for any g ∈ Fa,b we have that g−1x is an infinite word but may notbe reduced. The Fa,b action is defined as g−1 ·x = y where y is the infinite reducedform of g−1x. The Fa,b action does not preserve the Markov measure ν associatedto R = 1/3 but it does preserve the measure class. This gives rise to a (unitariz-

able) representation of Fa,b in L2(Σ+, ν). Set c(g, x) = dg∗νdν (x); this can easily be

computed as c(g, x) = 3−q(g,x) where q(g, x) = |g| if g−1x is already reduced, and

48 RHIANNON DOUGALL

q(g, x) = |g| − 2(g−1 ∧ x) in general. To see this it is enough to check cylinders: form > n, u = u1 · · ·um and g1 · · · gn, we have g−1u = g−1

n · · · g−1k+1uk+1 · · ·um so that∫

1[u](y)dg∗ν(y) =

∫1[u](y)dν(g−1 · y)

=

∫1[u](g · x)dν(x) =

∫1[g−1u](x)dν(x)

=1

43−|u|+k−n+k

The conclusion follows upon observing that g−1 ∧ u = k and |g| = n. Denote π1/2

for the unitary representation

π1/2(g)F (x) = c(g, x)1/2F (g−1 · x).

Let us compute the matrix coefficient g 7→ 〈π1/2(g)1,1〉L2(Σ+,ν), where 1 denotesthe constant unit function. We have

〈π1/2(g)1,1〉L2(Σ+,ν) =

∫Σ+

c(g, x)1/2dν(x)

Write Ek for the set with q(g, x) = |g| − 2k; or equivalently x with g−1 ∧ x = k.The measure of Ek takes values:

(14.1) ν(Ek) =

34 , for k = 0;143−k+1 2

3 , for 0 < k < |g|143−|g|+1, for k = |g|.

Therefore

〈π1/2(g)1,1〉L2(Σ+,ν) =3

4

√3−|g|

+6

43−|g|+1

√3|g|

+

|g|−1∑k=1

√3−|g|+2k 1

23−k

=3

2

√3−|g|

+1

2

√3−|g|

|g|∑k=1

√3

2k3−k

=3

2

√3−|g|

+1

2

√3−|g|

(|g| − 1)

=√

3−|g|

(3

2+

1

2(|g| − 1)

)Then g 7→ 〈π1/2(g)1,1〉L2(Σ+,ν) is identically g 7→ (1 + |g|

2 )√

3−|g|

.Spherical functions for the free group are explored in more detail by Figa-

Talamanca and Picardello [11].

15. The slowly increasing function

A formula for the slowly increasing function is given in [6]. We present the detailsneeded to verify it works.

Proposition 15.1. For any real series η(t) =∑∞n=1 t

−nBn there is a slowly in-creasing function c : N → ∞ so that ηc(t) =

∑∞n=1 t

−ncnBn has ηc(γ) = ∞ at

γ = lim supB1/nn and satisfies

cn+k ≤ cnck.

Proof. By hypothesis,

lim supn→∞

B1/nn = γ,

∞∑n=1

γ−nBn <∞.

First shift so thatDn = γ−nBn


and now the problem is

(15.1) lim supn→∞

D1/nn = 1,

∞∑n=1

Dn <∞.

Certainly Dn are decreasing (in order to have convergence in Eq 15.1) so D−1n are

increasing. A naive idea would be to set cn = D−1n so that every summand is 1!

(Far too optimistic!) A slowly increasing function necessarilty has lim sup c1/nn = 1,

but this is only sufficient, and so equation 15.1 is not enough. Failing this we mightthink to choose cn = D−1

q(n) with q(n) ≤ n. In order to have

∞∑n=1

DnD−1q(n) =∞

it is sufficient that the equation q(n) = n is satisfied for infinitely many n (no matterhow sparse the subset may be!). For instance if q(n) were constant in the range[N,N +M ] then certainly

D−1q(n+k) ≤ D

−1q(n) ≤ D

−1q(n)(1 + ε(k)).

provided n ∈ [N,N +M ], n+ k ∈ [N,N +M ]. (But this is too optimistic!)The solution is to take a sparse set (a collection Nr having NrN

−1r+1 → 0) and

“linearly interpolate” in such a way to make the extension of cNr = D−1Nr

slowlyincreasing. For n ∈ [Nr, Nr+1], set αr(n) ∈ [0, 1],

αr(n) =Nr+1 − nNr+1 −Nr

.

(So αr(Nr) = 1, αr(Nr+1) = 0.) Set

cn =(D−1Nr

)nαr(n)Nr

(D−1Nr+1

)n(1−αr(n))Nr+1

.

For brevity write d(r) = N−1r logD−1

Nr, whence

1

nlog cn = αr(n)d(r) + (1− αr(n))d(r + 1).

We may assume r are chosen with D−1Nr

> 1 (so d(r) > 0) and with N−1r logD−1

Nrmonotonically decreasing to 0 (so 0 < d(r + 1) < d(r)). It is immediate that1n log cn → 0.

We check the slowly increasing condition. Let k be arbitrary. We must showthat

limn→∞

log cn − log cn−k = 0.

Equivalently, that

limn→∞

log cn −n

n− klog cn−k = 0.

If n, n− k ∈ [Nr, Nr+1] then

1

nlog cn −

1

n− klog cn−k = [αr(n)− αr(n− k)]d(r)− (αr(n)− αr(n− k))d(r + 1).

When n is large enough (so r, Nr are large enough) we have

1

nlog cn−

1

n− klog cn−k] = [αr(n)−αr(n−k)]ε−[αr(n)−αr(n−k)]ε−δ = −δ[αr(n)+αr(n−k)].

Note that

αr(n)− αr(n− k) =k

Nr+1 −Nr=

kNr+1

1− NrNr+1

=k

Nr+1(1− εr).

So| log cn −

n

n− klog cn−k| ≤ Nr+1δ[αr(n) + αr(n− k)] ≤ kδ(1− εr).

50 RHIANNON DOUGALL

The reason for interpolation is to cover the disjoint ranges. It is sufficient tocheck for k = 1. Now if n = Nr and n− 1 < Nr we have

c1/nn

c1/(n−1)n−1

=(D−1Nr−1

) 1Nr (

D−1Nr

) 1Nr .

So1

nlog cn −

1

n− 1log cn−1 =

1

Nrd(r) +

1

Nrd(r − 1).

Since d(r) tends to 0 the conclusion follows.Now that we have convinced ourselves of the divergence and slowly varying

property it still remains to check that products have cnck ≥ cn+k (log cn+k ≤log cn + log ck). We can write

log cn+k =

(n∑q=1

αr(n)d(r) + (1− αr(n))d(r − 1)

)+

(k∑q=1

αr(n)d(r) + (1− αr(n))d(r − 1)

).

Since d(r) > 0 are decreasing in r, any linear combination has

td(r − 2) + (1− t)d(r − 1) > sd(r − 1) + (1− s)d(r),

giving the conclusion when n + k ∈ [Nr, Nr+1] and n, k < Nr. We also havemonotonicity, If n, n+ k ∈ [Nr, Nr+1] then αr(n) ≤ αr(n+ k) whence αr(n)d(r) +(1− αr(n))d(r − 1) < αr(n+ k)d(r) + (1− αr(n+ k))d(r − 1). �

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Department of Mathematical Sciences, Durham University, Upper Mountjoy, DurhamDH1 3LE

Email address: [email protected]

arxiv:2110.02147v1 [math.ds] 5 oct 2021

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