Harmonic Analysis, Ergodic Theory and Probability a conference in honor of

Izzy Katznelson's 70th Birthday

Partially supported by Stanford University, the Clay Mathematics Institute and the National Science Foundation

Stanford University, December 12-14, 2004

Since the early days of ergodic theory and the pioneering work von Neumann, ergodic theory and harmonic analysis have been intimately connected, often in surprising ways. A classic example of this is the use of harmonic analysis to prove convergence results for ergodic averages. Recently, there has been a great deal of interplay between these two fields, including work motivated by applications to Ramsey Theory. This interplay also touches upon subjects of active interest in probability theory, such as martingales, Bernoulli convolutions and random polynomials, as well as the Kakeya problem which is one of the outstanding open problems in harmonic analysis. This conference will bring together many of the leading mathematicians in these related areas to report on recent developments of broad interest and to point the way for exciting directions for future research. In this way we plan to honor the significant contributions of Izzy Katznelson in these areas on the occasion of his 70th Birthday.

OPEN PROBLEM SPEAKERS

SCHEDULE & ABSTRACTS Vitaly Bergelson (Ohio State University)

PARTICIPANTS Jean Bourgain (Institute for Advanced Study)

PHOTOS

Hillel Furstenberg (Hebrew University)

Ben Green (University of British Columbia)

Jean-Pierre Kahane (University of Paris, Orsay)

CONFERENCE LOCATIONS Bryna Kra (Penn State University & Northwestern)

The conference will be held in Jordan Hall, Room 040 Elon Lindenstrauss (NYU & Clay Mathematics Institute)

Assaf Naor (Microsoft Research)

TOPICS Yuval Peres (University of California, Berkeley)

Szemeredi's theorem and its generalizations Wilhelm Schlag (Caltech)

Multiple recurrence Mikhail Sodin (Tel Aviv University)

The Kakeya problem Terence Tao (University of California, Los Angeles)

Properties of random functions and polynomials Benjamin Weiss (Hebrew University)

ORGANIZERS CONTACT

Amir Dembo (Stanford University) Conference Coordination - Pat Cahill

Bryna Kra (Penn State University & Northwestern) Website - John Esposito

Elon Lindenstrauss (NYU & Clay Mathematics Institute)

OPEN PROBLEMS (gathered by N. Berger, A. Bufetov, C. Demeter and N. Frantzik-inakis)

I. Assani. Let (X,B, µ) be a finite measure space,Ti be measure preserving transforma-tions acting onX, andfi be bounded functions,i = 1, . . . , 2k − 1. The averages along thecubes for three terms(k = 2) and seven terms(k = 3) are defined as follows:

MN (f1, f2, f3)(x) =1N2

N∑m,n=1

f1(Tm1 x) · f2(Tn

2 x) · f3(Tm+n3 x)

and

MN (f1, f2, . . . , f7)(x) =1N3

N∑m,n,p=1

f1(Tm1 x) · f2(Tn

2 x) · f3(Tp3 x)·

f4(Tm+n4 x) · f5(Tm+p

5 x) · f6(Tn+p6 x) · f7(Tm+n+p

7 x).

Similarly, we define the averages along the cubes for2k − 1 terms and we denote them byMN (f1, f2, . . . , f2k−1). The averages along cubes for a single transformation (i.e.Ti = Tfor all i) were introduced by V. Bergelson who provedL2 convergence fork = 2. Hisresult was extended by B. Host and B. Kra who provedL2 convergence fork > 2 (for asingle transformation). I. Assani proved a.e. convergence in this case.

Problem 1. Do the averagesMN (f1, f2, . . . , f2k−1) converge inL2 and a.e. for everyk ∈ N?

Assuming that the transformationsTi are ergodic and commute, I. Assani remarks thatthe answer is yes for everyk, and without further assumptions the answer is yes fork = 2.If the transformations are weak mixing (not necessarily commuting) then the limit (a.e.and inL2) exist and equals the product of the integrals.

Problem 2. Find characteristic factors forL2 and a.e. convergence for the averagesMN (f1, f2, . . . , f2k−1).

J. Campbell. For a sequenceTn of operators andn1 < n2 < . . . < nk < . . . definethe oscillation operator

O2(Tn, f, nk)(x) :=

( ∞∑k=1

supnk<n≤nk+1

|Tnkf(x)− Tnf(x)|2

) 12

,

and thep-variation operator

Vp(Tn, f)(x) = supnk

(∑k

|Tnkf(x)− Tnk+1f(x)|p

) 1p

,

p > 2.An easy exercise shows that ifO2(Tn, f)(x) <∞ a.e. x for eachnk thenlimn→∞ Tnf(x)

exists a.e.Set now

T θnf(x) =

∑0<|k|≤n

eikθf(T kx)k

for eachθ ∈ [0, 1) andf measurable in the dynamical system(X,Σ,m, T ).1

2

Problem 3. The fundamental question is whether it is true that there exists a Wiener-Wintner result for these averages, i.e. whether for eachf ∈ L∞(X) there existsX0 ⊂ Xwith m(X0) = 1 such thatlimn→∞ T θ

nf(x) exists for eachx ∈ X0 andθ ∈ [0, 1).

Problem 4. If one wants to use harmonic analysis to answer this question, one approach isto prove an oscillation inequality for the continuous model and then transfer it back to theergodic theory setting. This amounts to proving that for each decreasing sequencetk ofpositive real numbers, one has

‖ supθ∈[0,1)

(∑k

suptk+1≤t<tk

∣∣∣∣∣∫

tk+1≤|y|<t

f(x− y)y

eiyθdy

∣∣∣∣∣) 1

2

‖2,∞ . ‖f‖2,

for f ∈ L2(R).

Denote also forf ∈ L1([0, 2π))

Cnf(x) =∑|j|≤n

f(j)eijx

Problem 5. Is it true that

Vp(Cn, f) : L2 → L2

boundedly?

H. Furstenberg. A system(X,σ, τ) is a compact metric spaceX together with two com-muting continuous transformationsσ, τ acting onX. A system(X,σ, τ) satisfies thedi-mension subadditivityproperty if for everyx ∈ X we have

dim σmτnxm,n∈N ≤ dim σmxm∈N + dim τnxn∈N

wheredim denotes the Hausdorff dimension. It satisfies thetransversality property ifwheneverA,B ⊂ X are closed such thatσA ⊂ A, τB ⊂ B, we have

dim(A ∩B) ≤ maxdimA+ dimB − dimX, 0.

Problem 6. Let n ∈ N andσ, τ be expanding (continuous) homomorphisms ofTn withthe standard metric such that every orbitσmτnxm,n∈N is either finite or dense,x ∈ Tn.Does the system(X,σ, τ) satisfy the dimension subadditivity and/or the transversalityproperty?

The assumption that bothσ andτ are expanding is necessary as one can see by choosing

σ =(

2 00 2

), τ =

(2 10 2

); then it can be shown that the system(T2, σ, τ) does not

satisfy the dimension subadditivity or the transversality property.

Problem 7. Suppose thatΛ is a finite additive group with the discrete metric. On thesequence spaceX = ΛN with the product metric we letσ be the shift transformation andτ = 1 + σ (i.e. (τx)i = xi + xi+1). Does the system(X,σ, τ) satisfy the dimensionsubadditivity property?

Note that this system doesn’t satisfy the transversality property because there are manyinfinite closed subsets invariant under both transformations.

3

S. Gunturk. [Invariant sets of a class of piecewise affine maps on the Euclidean space]Let L be then× n lower triangular matrix of1’s, i.e.,

L =

1 0 · · · 0...

......

......

... 01 · · · · · · 1

,

ande be then dimensional vector(1, · · · , 1)>.Let x be a real number,Π = Ω1, . . . ,ΩK be a partition ofRn into Lebesgue measur-

able sets, andD = d1, . . . ,dK be a set of vectors inZn. Given the triple(x,Π,D), wedefine a piecewise affine mapT : Rn → Rn by

T (u) = Lu + xe + di, if u ∈ Ωi.

We say thatT is stable if there exists a bounded setA ⊂ Rn of positive Lebesguemeasure such thatT (A) ⊂ A. It is easy to see thatT is stable if and only if there existbounded setsB andC in Rn of positive measure such that

∞⋃k=0

T k(B) ⊂ C.

We would like to understand the invariant sets of the mapT , given the triple(x,Π,D).The following theorem is not difficult to prove:

Theorem1 Let x be an irrational number andT be stable. Then there exist a finite andnon-empty collection of disjoint setsτ1, . . . , τN ⊂ Rn such that

(a) eachτi tilesRn by Zn translations, i.e., the collectionτi + m : m ∈ Zn

is a partition ofRn, and

(b) if we letΓ =⋃τi, thenT (Γ) = Γ. Also, we haveΓ =

∞⋂k=0

T k(A).

Notes.(1) Note that the affine mapu 7→ Lu + xe is always unstable in all dimensions. The

role of the partitionΠ and the corresponding set of translationsD is to overcomethis inherent instability.

(2) Perhaps the simplest stable mapsT are those given (implicitly) byT (u) = Lu +xe (mod 1). Here, we have the invariant setΓ = [0, 1)n. (In reality, the corre-sponding partitionΠ would have infinitely many sets, but one can always reduceit to a finite partition by restricting thedi to those produced byu ∈ [0, 1)n. Ingeneral, this requiresK = 2n.)

(3) The most interesting case is whenK = 2 regardless of the dimensionn, i.e., whenthere are only two sets in the partitionΠ. This is the most challenging setup forfinding stable mapsT , though it is known that stable maps exist in all dimensions(for values ofx in an interval).

1For a proof, see S. Gunturk, N.T. Thao, “Ergodic Dynamics in Sigma-Delta Quantization: Tiling In-variant Sets and Spectral Analysis of Error,” Advances in Applied Mathematics, in press. Available athttp://www.cims.nyu.edu/ ∼gunturk/research.html

4

(4) In one dimension, there is a strong link to interval exchange transformations.Higher dimensional versions are not much studied.

Some open problems.

Problem 8. For which(x,Π,D) is T stable?(Here, it would be desirable to have a characterization, or a set of sufficient or necessaryconditions that can be checked, preferably via a finite algorithmic test.)

Problem 9. Determine the (maximal) invariant setΓ as an explicit function of the param-eters ofT (i.e.,(x,Π,D)).

Problem 10. If this is not possible, predict the size ofΓ and/or the number of tilesN inΓ. (Here, the single tile caseN = 1 is particularly important as then the dynamics ofTwithin Γ is isomorphic to the skew translationu 7→ Lu + xe (mod1) on [0, 1)n, which isuniquely ergodic for irrational values ofx.)

Problem 11. What is the regularity ofΓ for a given mapT?(For instance, determine the Hausdorff dimension of∂Γ. These results are useful in deriv-ing quantitative results for the convergence of ergodic averages via the standard machineryof Weyl sums.)Tiling invariant sets are observed even whenx is rational, however the theorem mentionedabove is unable to explain this. Explain the tiling phenomenon without using ergodictheory arguments.

J. P. Kahane. [Genericity and Prevalence]Given a class of functions, if we say that a property holdsin general, what does it mean?We restrict ourselves to classes of functions that are Frechet spaces. A “property” can

be identified with a subset of the space.Since a Frechet space is a Baire space, we have the notion ofgeneric property,mbelongs

to this class or this intersection. eaning that it holds on a countable intersection of denseopen sets. Instead ofgenericwe also sayquasi sure.

There is also the notion ofalmost sure, when we equip the Frechet space with a proba-bility measure (we always assume that it is carried by aσ-compact set in the space).

When the Frechet space is given, the notion of generic, or quasi sure, is well defined,but the notion of almost sure depends on the choice of a probability.

However, using the group structure of the space, we’ll say that a property isprevalentifit is almost sure forsomeprobability measure andall its translates.

The notion was introduced by Christiansen in 1972, and was developed by Hunt, Sauer,and Yorke in 1992 (Bull. AMS 27, 217–238; see also Bull. AMS 28, 306–307). I learnedit from Stephane Jaffard and his student Aurelia Fraysse, who have a series of examplesin various spaces, where the properties they consider (multifractal formalism, failure ofanalytic continuation) are both generic and prevalent (2005). A simple example inC(R)is the nowhere-differentiability, known to be generic and also prevalent, using the Wienermeasure (Holicky and Zagicek, Acta Univ. Carol 41 (2000), 7–11).

Our general question can be split into two parts:

Problem 12. Give other examples of spaces and properties both generic and prevalent.

Problem 13. Give spaces and properties which are as different as possible from the genericand from the prevalent point of view.

Here is a contribution for question 2): The Frechet space is the real spaceC(R), andwe consider a Cantor setE in R and a continuous probability measureµ onR. We look at

5

properties of the images ofE and ofµ by f ∈ C(R), say,F = f(E) andν = µ f−1.Generically,F is a Kronecker set, meaning a Cantor set such that each function continuousonF whose absolute value is 1, can be approximated uniformly by imaginary exponentials.It is a thin set in most aspects of harmonic analysis (R. Kaufman, 1967).

Using this fact, one sees thatν is generically a singular measure, and moreover

lim supn→∞

|ν(n)| = 1.

The prevalent properties are quite different. Prevalently,F has non-empty interior(question: is it a union of intervals?). Prevalently,ν is absolutely continuous, its den-sity is C∞ and moreover, given any non quasi-analytic class of infinitely differentiablefunctions (or a countable intersection of such classes), its density belongs to this class orthis intersection.

Y. Katznelson. A setΛ ⊂ N is a set of recurrence for the system(X, d, T ) if for everyopen setU ⊂ X there existsn ∈ Λ such thatU ∩ T−nU is nonempty. Λ is a set oftopological recurrence if it is a set of recurrence for every minimal topological system,and a set ofBohr recurrence if it is a set of recurrence for every translation on a finitedimensional torus (with the standard metric).

Problem 14. If Λ ⊂ N is a set of Bohr recurrence is it necessarily a set of topologicalrecurrence?

For background see Y. Katznelson,Chromatic Numbers of Caley Graphs onZ andrecurrence, Combinatorica, 21, No 2, 211-219, also available at

http://math.stanford.edu/ ∼katznel

B. Kra.

Problem 15. Let (X,B, µ) be a finite measure space and letT1, . . . , Tk be ergodic com-muting measure preserving transformations acting onX. If ε > 0 when is the set

n ∈ N : µ(A ∩ T−n1 A ∩ . . . ∩ T−n

k A) > µ(A)k+1 − ε

syndetic?

If T1 = T, T2 = T 2, . . . , Tk = T k, V. Bergelson, B. Host and B. Kra showed that theset is always syndetic fork ≤ 3 and gave examples where it is empty fork = 4.

Problem 16. Let (X,B, µ) be a finite measure space,T1, . . . , Tk be commuting measurepreserving transformations acting onX, andp1, . . . , pk be linearly independent integervalued polynomials with zero constant term. Ifε > 0 when is the set

n ∈ N : µ(A ∩ T−p1(n)1 A ∩ . . . ∩ T−pk(n)

k A) > µ(A)k+1 − ε

syndetic?

For a single transformation (i.e.Ti = T for all i) N. Frantzikinakis and B. Kra showedthat the set is syndetic for everyk ∈ N.

6

E. Lesigne. Two measure-preserving dynamical systems(X,T, µ), (Y, S, ν) are said tobe weakly disjointif, given f ∈ L2(X,µ), g ∈ L2(Y, ν), there exist setsA ⊂ X andB ⊂ Y , µ(A) = 1, ν(B) = 1, such that for anyx ∈ A, y ∈ B, the sequence

1N

N−1∑n=0

f(Tnx)g(Sny)

converges asN →∞.We recall the following properties: disjointness implies weak disjointness; a dynamical

system with discrete spectrum is weakly disjoint from any other system; the Chacon mapis weakly disjoint from any other system (in particular, from itself); the Morse system isweakly disjoint from any ergodic system; two systems of positive entropy are never weaklydisjoint.

Problem 17. Assume that a measure-preserving dynamical system is weakly disjoint fromany ergodic system. Does it follow that it is disjoint from any measure-preserving dynam-ical system?

A subsetE ⊂ Z is called aset of strong recurrencefor a measure-preserving dynamicalsystem(X,µ, T ) if for an arbitrary subsetA ⊂ X of positive measure, there existsε > 0such thatµ(A ∩ T−nA) > ε for infinitely manyn ∈ E.

Problem 18. LetE be a set of strong recurrence for an arbitrary ergodic system. Does itfollow thatE is a set of strong recurrence for any measure-preserving dynamical system.

A. Naor. LetX,Y be metric spaces. Assume that there exists a constantC such that forany subsetZ ⊂ X, anyK > 0, and any Lipschitz mapF : Z → Y with Lipshitz constantK, one can find an extensionF : X → Y , agreeing withF on Z and having Lipschitzconstant at mostCK. The smallest suchC is denoted bye(X,Y ) (we sete(X,Y ) = ∞if no such constant exists).

The classical theorem of Kirszbraun (1934) says thate(H1,H2) = 1 for two HilbertspacesH1,H2: in other words, a Lipschitz map from a subset of a Hilbert space intoanother Hilbert space can be extended to a global map between these spaces without anyincrease in the Lipschitz constant.

In 1992, Keith Ball showed thate(L2, Lp) is finite for1 < p < 2.

Problem 19. Is e(L2, L1) finite or infinite?

For maps into Banach spaces, a different approach to the extension problem was sug-gested by Lee and Naor (2003). Given a metric spaceX, assume that there exists a constantC such that for any metric spaceY , containingX, any Banach spaceZ, and any Lipschitzmapf : X → Z with Lipschitz constantK, there exists an extensionF : Y → Z, agree-ing with f onX and having Lipschitz constant at mostCK. The smallest suchC is calledtheabsolute extendability constantofX and denoted byae(X) (again, we setae(X) = ∞if no such constant exists). The problem of estimating the absolute extendability constantis already interesting for finite metric spaces. It is known that there exist two constantsC1

andC2 such that

C1

√log n

log log n≤ sup|X|≤n

ae(X) ≤ C2log n

log log n.

Problem 20. What is the precise asymtotics inn of sup|X|≤n ae(X)?

7

D. Ornstein. Find an approach to KAM (i.e. a method to produce invariant curves andsurfaces) that is: (i) elementary, (ii) unified, (iii) gives results that are optimal, (iv) relaxesthe condition: small perturbation of a completely integrable system. This can be done indimensions1 and2 (work partly joint with Y. Katznelson).

In dimension1, given a diffeomorphismψ of the circle the method gives the completeanswer to the smoothness of the invariant measure (i.e. the function conjugatingψ to arigid rotation) in terms of the smoothness ofψ and the rate of growth of the coefficients ofthe continued fraction expansion of the rotation numberα. It gives the optimal results forany diophantineα (completing results of Herman and Yoccoz).

The main results in dimension2 are:(A) Let ψ be a measure preserving diffeomorphism of the disc and assume that the

rotation numberα of ψ restricted to the boundary is diophantine, and thatψ is smoothenough givenα. Then the method produces invariant curves of optimal smoothness. Forexample ifψ ∈ C3+γ and the coefficients of the continued fraction expansion ofα growpolynomially then there are invariant curves (filling a set of positive measure near theboundary) inC2+γ . A recent result of Herman produces invariant curves inC1+γ . Heuses standard KAM where the loss in smoothness is not optimal because the curve and itsinvariant measure are produced together (and the invariant measure is less smooth than theinvariant curve). Standard KAM can be applied because if the boundary is rotated rigidlythen near the boundary we have a small perturbation of a completely integrable system.Conjugating to this situation requires more smoothness forψ. Note that we could replacethe circle by any invariant curve inC1+γ .

Problem 21. Is there a3-dimensional analog of (A)?

(B) We assume thatψ is a twist map of the disc or the cylinder. Then a necessary andsufficient condition for the existence of an invariant curve inCβ (β > 2) with rotationnumberα is: A “sufficiently long” finite orbit lies on aCβ curve. We get the optimalsmoothness needed forψ in terms ofα andβ and “sufficiently long” depends onα andβ.The completely integrable system has disappeared (as in the Morse twist) but the ghost ofa “small” perturbation appears in “sufficiently long”. The main machinery of our methoddoes not depend on being in dimension1 or 2. We can get an analog of (B) for invariant2-tori in a3-torus.

Problem 22. Find and prove optimal smoothness conditions in this situation.

Y. Peres. [ Projections of planar Cantor sets and a related Kakeya set] LetKn be theproduct of twon-stage middle-half Cantor sets, and letK = ∩Kn be the product of thetwo Cantor sets. Forθ ∈ [0, π), let Pθ be the operator of projection at angleθ. It iswell-known (Besicovitch) that

limn→∞

∫ π

0

µ(Pθ(Kn))dθ =∫ π

0

µ(Pθ(K))dθ = 0.

Problem 23. What is the rate of convergence? It is known (Peres, Solomyak) that

C

n<

∫ π

0

µ(Pθ(Kn))dθ <c

log∗(n)

for some constantsC andc, wherelog∗(n) is defined to be the minimalk such thatn ≤exp(k)(1) andexp(k) denotes thekth iterate ofexp. .

Note that this is related to a construction of Kakeya set, i.e. a compact set of area zerocontaining a unit interval in every direction: LetK be a middle half cantor set, and let

8

K(1),K(2) ⊆ R2 beK(1) = K × 0 andK(2) = 12K × 1. TakeC = αx + (1 −

αy)|0 < α < 1, x ∈ K(1), y ∈ K(2). ThenC contains a unit interval in every directionwithin a range of 60 degrees, and the cutsC ∩ (α × R) are projections ofK2.

D. Rudolph. We start by defining anRn or Zn borel foliation of a Polish spaceX. Thiswill be a countable collection of chartsCi which have the formSi×Bri

whereSi is PolishandBri is a ball or box of radiusri centered at~0 in Rn or Zn. With each chartCi we havea borel injectionφi : Ci → X, i.e.

(1) the setsφi(Ci) cover(2) on intersectionsIi,j = φ−1

i (φi(Ci)∩φj(Cj)) restricted to to a leaf(s×Bri)∩Ii,j

will have the forms× (Bri ∩ (fi,j,s(Brj ))),

wherefi,j,s is an isometry ofRn or Zn

(3) for all x ∈ X, x ∈ φi(Ci) if one seeks to develope the leaf throughx starting withs × Bri

, φi(s,~v) = x, and extending through leaves of charts that intersect thisleaf, one obtains a full copy ofRn or Zn.

We call the leaf throughx, Lx. A natural example would be free borel actions ofRn or Zn

on a Polich space.Now supposeµ is a borel probability measure onX. On any chartCi we can consider

µi = (φ−1i )∗(µ), the pull back ofµ to a chart. On a chart one can take the Rohklin decom-

position of the measureµi =∫µsdµi(s). The measuresµs are unique up to normalization.

This means that on intersections of charts, the measuresµi and(φ−1i φj)∗µj agree up to

a normalization. This means, forµ a.e. x one can construct a measureµx obtained byextending the measures chart by chart with the correct normalization, to the full leafLx.For expliciteness we normalize so that for a.e.x µx(B1(x)) = 1, whereB1(x) is the unitball aboutx in Lx.

For anyf ∈ L1(µ) one can use theµx to compute leaf averages

Ar,µ(x) =∫fdµx

µx(Br(x)).

Problem 24. Is it true, at this level of generality thatAr,µ(x) converges inr a.s., or inalternatively inL1(µ)?

It is known that (1) the answer is yes forR or Z foliations, by a slight extension of theHurewicz ergodic theorem. (2) ForRn or Zn foliations one does have a general maximalinequality (these are joint work with E. Lindenstauss).

W. Schlag. Let Hω,λ be the disctete quasi-periodic Schrodinger operator given by theskew-shift:

(Hω,λψ)n = −ψn+1 − ψn−1 + λv(Tnω (x, y))ψn,

where(ψn) ∈ l2, v(x, y) = cos(2πx) for each(x, y) ∈ T2 andTω(x, y) = (x+ y, y+ω)is the skew-shift on the 2 dimentional torusT2. For each eigenvalueE we denote by

Aj(x, y;E) =(

λv(T jω(x, y))− E −1

1 0

),

Mn(x, y;E) =1∏

j=n

Aj(x, y;E),

Ln(E) =∫

T2

1n

log ‖Mn(x, y;E)‖dxdy

9

andL(E) = limn→∞ Ln(E) = inf Ln(E) denotes the Lyapunov exponent. ClearlyL(E) ≥ 0, for all E andλ. J. Bourgain, M. Goldstein and W. Schlag have proved that foreachε there exists a setΩε ⊂ T with mes[T \ Ωε] < ε and a large constantλ0(ε) suchthat for eachω ∈ Ωε andλ ≥ λ0, the Lyapunov exponentsL(E) are strictly positive forall energies, for a.e.(x, y).

Problem 25. Is it true that one can replaceλ ≥ λ0 with λ > 0 in the above?

This would contrast with the case of simple shift onT , Tω(x) = x+ω, where the resultonly holds forλ > 2.

Scott Sheffield. [Differentiability of infinity harmonic functions]We begin with an× n-grid on the plane and a real-valued functionf(x, y), defined on

the vertices of the grid.The game oftug of warhas two players and a moving point, which is placed at the

origin in the beginning of the game. Each player moves the point into an adjacent vertexof the grid. When the point hits the boundary of the grid at a point(x0, y0), the first playercollectsf(x0, y0) dollars.

Problem 26. Is it true that there existN ∈ N andδ > 0 such that ifn > N and|f(x, y)−y| < δn, then the optimal first move for the first player is up?

We can now consider the continuous version of the game. Heref is a continuous func-tion on the boundary of then × n-grid, and the players are allowed to move the point atdistance one in any direction. FixC > 0. We say that the point goes “up” by a given moveof the game if the vertical coordinate of the vector by which it is moved is at leastC. Itwould be interesting to answer the question in this case also.

B. Solomyak. [Interior points of self-similar sets]A nonempty compact setE ⊂ R is calledself-similar if there existsm ∈ Z, m ≥ 2,

λ1, . . . , λm ∈ R, 0 < λi < 1, andd1, . . . , dm ∈ R such that

E = ∪mi=1fi(E), where fi(x) = λix+ di.

We shall sometimes write

E = E(λ1, . . . , λm; d1, . . . , dm).

Problem 27. LetE be a self-similar of positive Lebesgue measure. DoesE have nonemptyinterior?

If m = 2, then it is easy to see that the answer is ”yes”; we shall therefore assumem ≥ 3 in what follows.

Thesimilarity dimensionof E is α > 0 such thatm∑

i=1

λαi = 1.

It is well-known that the Hausdorff dimension ofE does not exceed its similarity di-mension, so we are interested in the caseα ≥ 1. If α = 1, then the answer to Question27 is ”yes” (A. Schief). Forα > 1, it is known (Marstrand and Mattila), that, givenλ1, . . . , λm, for almost everym-tuple d1, . . . , dm the corresponding self-similar set haspositive Lebesgue measure. We thus have

Problem 28. Take a vector(λ1, . . . , λm), whose self-similarity dimension is greater than1. Is it true that for almost every (with respect to Lebesgue) vector(d1, . . . , dm), the setE = E(λ1, . . . , λm; d1, . . . , dm) has nonempty interior?

10

Questions 27 and 28 are already interesting for special families of self-similar sets. Forexample, let14 < λ < 1

2 , denote byCλ the standard middle1−2λ Cantor set and considerits Cartesian squareKλ = Cλ × Cλ. For 0 ≤ θ ≤ 2π, denote bypθ the orthogonalprojection onto a line with slopeθ. Consider the familypθ(Kλ). By Marstrand, for almostall θ, the Lebesgue measure ofpθ(Kλ) is positive. Is it also true that the interior ofpθ(Kλ)is nonempty for almost allθ? is there at least a singleθ such thatpθ(Kλ) has positivemeasure but empty interior?

One may also ask what happens in higher dimensions. Recently, an example was found(joint with M. Csornyei, T. Jordan, M. Pollicott, and D. Preiss) of a self-similar set inR2

with positive Lebesgue measure but empty interior.

B. Weiss. [ Shanon Entropy of linear factors]

Problem 29. Let Xn∞n=−∞ be a stationary process in the reals such thath(Xn) = ∞.Assume further that theXn-s are bounded. Let0 6= cn ∈ `1(Z) and letYn be theconvolution

Yn =∞∑

k=−∞

Yn−kck.

Is it true thath(Yn) = ∞?

Problem 30. SupposeXi,j(i,j)∈Z2 is a stationary finite valued process. Let0 6= cn ∈`1(Z2) and letY = c ∗X. Is it true thath(X) = h(Y )?

M. Wierdl. Let σnn∈N be a non-increasing sequence satisfying0 ≤ σn ≤ 1, andXnn∈N be a sequence of independent0− 1 valued random variables withP (Xn(ω) =1) = σn. Givenω ∈ Ω we construct the integer subsetAω of N by takingk ∈ Aω if andonly if Xk(ω) = 1. By writing the elements ofAω in increasing order we get a sequencean(ω)n∈N. M. Boshernitzan and J. Bourgain showed that if

(1) limt→∞

w(t)log t

= ∞

wherew(t) =∑

n≤t σn thenω-almost surely we have: in every measure preserving sys-tem the ”random averages”

1w(t)

∑n≤t

n∈Aω

Tnf

converge inL2 ast→∞. Moreover, they showed thatω-almost surely the setAω is a setof (single) recurrence.

Problem 31. Assuming that (1) holds, is it true thatω-almost surely we have: for everymeasure preserving system and bounded measurable functionsf, g the multiple ergodicaverages

1w(t)

∑n≤t

n∈Aω

Tnf T 2ng

converge inL2 ast→∞?

A set Λ ⊂ N is called a set ofdouble recurrence if for every measure preservingsystem and measurable setA with positive measure there existsn ∈ Λ such thatµ(A ∩T−nA ∩ T−2nA) > 0.

11

Problem 32. Assuming that (1) holds, is it true thatω-almost surely the setAω is a set ofdouble recurrence?

M. Wierdl notes that the answer to both questions is yes ifσn = 1/na for some0 ≤a < 1/2, and unknown ifa = 1/2.

Schedule

Sunday, December 12

9:15 - 9:45 Coffee and Registration

9:45 - 10:00 Opening remarks

10:00 - 10:45 Jean Bourgain - Harmonic analysis aspects of Ginzburg Landau minimizers

11:00 - 11:45 Wilhelm Schlag - On ergodic matrix products and fine properties of the eigenfunctions of the associated difference equations

12:00 - 2:00 Lunch break

2:00 - 2:45 Ben Green - Progressions of length four in finite fields

3:00 - 3:30 Coffee and refreshments

3:30 - 4:15 Terence Tao - Quantitative ergodic theory

4:30 - 5:20 Open problems session

Izzy Katznelson - Topological Recurrence and Bohr Recurrence Mate Wierdl - Multiple convergence and recurrence along random sequences James Campbell - Oscillation and variation estimates for the Carleson operator, and

rotated ergodic averages. Daniel Rudolph - Another Ergodic theorem, maybe Idris Assani - On the pointwise convergence of ergodic averages along cubes for not

necessarily commuting measure preservng transformations Yuval Peres - Projections of planar Cantor sets and a related Kakeya set

Monday, December 13

9:30 - 10:15 Mikhail Sodin - Zeroes of Gaussian Analytic Functions

10:30 - 11:00 Coffee and refreshments

11:00 - 11:45 Yuval Peres - Determinantal Processes And The IID Gaussian Power Series

12:00 - 2:00 Lunch break

2:00 - 2:45 Assaf Naor - The Lipschitz Extension Problem

3:00 - 3:30 Coffee and refreshments

3:30 - 4:15 Elon Lindenstrauss - Invariant measures for partially isometric maps

4:30 - 5:20 Open problems session

Scott Sheffield - Infinity harmonic functions Boris Solomyak - Interior points of self-similar sets Emmanuel Lesigne - Two questions around ergodic disintegration Don Ornstein Sinan Gunturk - Invariant sets of a class of piecewise affine maps on Euclidean space

and/or other problems Jean-Pierre Kahane - genericity and prevalence

6:30 - 9:30 Banquet at the Faculty Club in honor of Y. Katznelson, with Bob Osserman as emcee, a historical talk by Jean-Pierre Kahane and a toast by Yonatan Katznelson.

Tuesday, December 14

9:30 - 10:15 Benjamin Weiss - On entropy of stochastic processes

10:30 - 11:00 Coffee and refreshments

11:00 - 11:45 Bryna Kra - Lower bounds for multiple ergodic averages

12:00 - 1:30 Lunch break

1:30 - 2:15 Vitaly Bergelson - IP versus Cesaro

2:30 - 3:15 Hillel Furstenberg - Hausdorff Dimension of Orbit Closures and Transversality of Fractals

3:30 - 4:00 Conference conclusion (Coffee and refreshments)

Abstracts

Vitaly Bergelson - Ohio State University Title: IP versus Cesaro Abstract: While traditional ergodic theory concerns itself with the study of the limiting behavior of various Cesaro averages, IP ergodic theory utilizes the notion of IP-convergence which is based on Hindman's finite sums theorem. This usually allows one to refine and enhance the results obtained via the Cesaro averages. An example of such an enhancement is provided by the Furstenberg-Katznelson IP Szemeredi theorem. We shall review some of recent developments in IP ergodic theory and formulate new interesting problems and conjectures.

Jean Bourgain - Institute for Advanced Study Title: Harmonic analysis aspects of Ginzburg Landau minimizers

Hillel Furstenberg - Hebrew University Title: Hausdorff Dimension of Orbit Closures and Transversality of Fractals Abstract: It sometimes happens that the dynamics of a commuting pair of transformations is more easily described than that of the individual transformations. We describe a situation of this type and the possibility that "large" orbits under the two transformations together and a "tiny" orbit under one of the transformations may imply a "large" orbit under the other. Here size is measured by Hausdorff dimension, and we are naturally led to problems regarding fractals.

Ben Green - University of British Columbia Title: Progressions of length four in finite fields Abstract: If G is a finite abelian group with size N, define r_4(G) to be the size of the largest subset of G which does not contain four distinct elements in arithmetic progression. Gowers showed that r_4(Z/NZ) is bounded above by N/(log log N)^c for some c > 0. I would like to discuss some joint work with Terry Tao in which we show that r_4(G) = N/(log N)^c for the particular group G = (Z/5Z)^n. The argument involves `quadratic fourier analysis' on `quadratic submanifolds': I will attempt to explain what that means. I will also discuss the prospects of generalising out result to arbitrary G.

Bryna Kra - Penn State University & Northwestern Title: Lower bounds for multiple ergodic averages Abstract: Recent developments in multiple ergodic averages have lead to new combinatorial consequences. I will discuss what happens to a set of integers with positive upper density when it is translated along certain sequences of integers and one takes the intersection of these sets. It turns out that substantially different behavior occurs if the sequence is formed using linear integer polynomials or formed using rationally independent integer polynomials. The first case corresponds to Szemeredi's Theorem for arithmetic progressions, and we have tight lower bounds on the size of the intersection for progressions of length 3 and 4, but no such bounds for longer progressions. For independent polynomials, such as polynomials of differing degrees, we have tight lower bounds on the size of every intersection.

Elon Lindenstrauss - NYU & Clay Mathematics Institute Title: Invariant measures for partially isometric maps Abstract: Consider an irreducible non-hyperbolic toral automorhpism on the four turus (which is the minimal dimension possible). This map contracts in one (one dimensional) direction, expands in another, and acts isometrically by rotations on the remaining two dimensions. This is prototypical to a larger class of maps which are partially isometric in the same sense. One invariant measure for this map is Lebesgue measure, which as an abstract measure preserving system has been shown by Katznelson to be Bernoulli, just like for the hyperbolic case. However if one consider the class of all invariant measures, there are nontrivial restrictions, and thare are many intriguing open questions.

As I will explain in my talk, classifying these invariant measures is closely related to Furstenberg's famous

conjecture about x2 x3 invariant measures on the circle R/Z.

Part of my talk will be based on joint work with Klaus Schmidt

Assaf Naor - Microsoft Research Title: The Lipschitz Extension Problem Abstract: The Lipschitz extension problem asks for conditions on a pair of metric spaces X,Y such that every Y-valued Lipschitz map on a subset of X can be extended to all of X with only a bounded multiplicative loss in the Lipschitz constant. This problem dates back to the work of Kirszbraun and Whitney in the 1930s, and has been extensively investigated in the past two decades. The methods used in this direction are based on geometric, analytic and probabilistic arguments. In particular, the methods involve stable processes, random projections, random partitions of unity and the analysis of Markov chains in metric spaces. In this talk we will present the main known results on the Lipschitz extension problem, as well as several recent breakthroughs.

Yuval Peres - University of California, Berkeley Title: Determinantal Processes And The IID Gaussian Power Series Abstract: Discrete and continuous point processes where the joint intensities are determinants arise in Combinatorics (noncolliding paths, random spanning trees) and Physics (Fermions, eigenvalues of Random matrices). For these processes the number of points in a region can be represented as a sum of independent, zero-one valued variables, one for each eigenvalue of the relevant operator. In recent work with B. Virag, we found that for the Gaussian power series with i.i.d. coefficients, the zeros form a determinantal process, governed by the Bergman Kernel. A partition identity of Euler, and a permanent-determinant identity of Borchardt (1855) appear in the proof. The determinantal description yields the exact distribution of the number of zeros in a disk. The process of zeros is invariant for a natural dynamics (we'll see a movie).

Wilhelm Schlag - Caltech Title: On ergodic matrix products and fine properties of the eigenfunctions of the associated difference equations Abstract: We will present some recent work on difference equations on the one-dimensional lattice. They are typically studied by means of transfer matrices, which define the associated co-cycles. Assuming positive Lyapunov exponents, we will discuss properties of the distribution of eigenvalues in the stochastic limit. This is joint work with Michael Goldstein.

Misha Sodin - Tel-Aviv University Title: Zeroes of Gaussian Analytic Functions Abstract: Geometrically, zeroes of a Gaussian analytic function are intersection points of an analytic curve in a Hilbert space with a randomly chosen hyperplane. Mathematical physics provides another interpretation as a gas of interacting particles. In the last decade, these interpretations influenced progress in understanding statistical patterns in the zeroes of Gaussian analytic functions, and led to the discovery of remarkable canonical models with invariant zero distribution. We shall discuss some of recent results in this area. The talk is based on joint works with Boris Tsirelson.

Terence Tao - UCLA Title: Quantitative ergodic theory Abstract: There are many techniques used in the study of multiple recurrence (or equivalently in detecting arithmetic progressions and similar objects). The combinatorial and Fourier-analytic approaches tend to work in "finitary" settings such as the cyclic group Z/NZ, whereas the ergodic theory approach works instead in the setting of an infinite measure-preserving system, with the two settings being linked via a transference principle which requires the axiom of choice. The ergodic theory methods are technically simpler (modulo standard machinery such as measure theory and conditional expectation) and are more easily applied to a wide range of problems, but the combinatorial and Fourier methods give more concrete bounds and can apply to certain settings (notably to subsets of sparse pseudorandom sets, of which the primes are a good example) for which there does not yet appear to be an infinitary analogue.

In this talk we discuss a compromise approach, which we dub "quantitative ergodic theory", in which we work in the finitary setting of Z/NZ but still exploit the philosophy and ideas from ergodic theory (e.g. sigma algebra factors, almost periodic functions, conditional expectation). This for instance allows one to give an elementary proof of Szemeredi's theorem based on ergodic methods (requiring no Fourier analysis, and no sophisticated

combinatorial tools otherthan van der Waerden's theorem) which also provides a quantitative (but rather poor) bound. This theory was also a crucial component of the recent result of Ben Green and the author that the primes contain arbitrarily long arithmetic progressions.

Benjamin Weiss - Hebrew University of Jerusalem Title: On entropy of stochastic processes Abstract: I will discuss some new observations on the relationship between the entropies of two stochastic processes, one of which is a linear factor of the other. The notion of "Finitely Observable" functions of processes will be defined and a remarkable new characterization of the entropy will be given.

Participants

Tarn Adams Stanford University Andres Angel Stanford University Idris Assani University of North Carolina, Chapel Hill Ioan Bejenaru University of California, Los Angeles Vitaly Bergelson Ohio State University Noam Berger Caltech Alexander Bufetov Princeton University Jean Bourgain Institute for Advanced Study Svetlana Butler University of Illinois, Urbana Champaign James T Campbell University of Memphis Chris Connell Indiana University David Damanik California Institute of Technology Manav Das University of Louisville Amir Dembo Stanford University Ciprian Demeter University of California, Los Angeles Iulia Demeter University of California, Los Angeles Yasha Eliashberg Stanford University Mehmet Burak Erdogan University of Illinois Jacob Feldman University of California, Berkeley Alexander Fish Hebrew University of Jerusalem Matt Foreman University of California, Irvine Nikos Frantzikinakis Penn State University Hillel Furstenberg Hebrew University of Jerusalem Michael Goldstein University of Toronto Ben Green Trinity College, Cambridge John Griesmer The Ohio State University Sinan Gunturk New York University J. Ben Hough University of California, Berkeley Michael Johnson Northwestern University Jean-Pierre Kahane University of Paris-Sud Soyoung Kang Purdue University Yitzhak Katznelson Stanford University Silvius Klein University of California, Los Angeles Wojciech Kosek Colorado College Bryna Kra Northwestern University Manjunath Krishnapur University of California, Berkeley Izabella Laba University of British Columbia David Lecomte Stanford University Doron Levy Stanford University Elon Lindenstrauss Princeton University Emmanuel Lesigne University of Tours Darwin R. Lo Stanford University Raul Lozano Cisco Systems, Inc. Brian Macdonald Johns Hopkins University Indur B. Mandhyan Carruth McGehee Louisiana State University Mark Meckes Stanford University Roman Muchnik University of Chicago

Asaf Nachmias University of California, Berkeley Assaf Naor Microsoft Research Don Ornstein Stanford University Robert Osserman Stanford University Yuval Peres University of California, Berkeley Gabor Pete University of California, Berkeley Rafal Pikula The Ohio State University Vidhu S. Prasad University of Massachusetts, Lowell Marina Ratner University of California, Berkeley Daniel J. Rudolph Colorado State University Roman Sasyk Purdue University Wilhelm Schlag Caltech Scott Sheffield University of California, Berkeley Pablo Shmerkin University of Washington Justin Conrad Sinz University of Chicago Boris Solomyak University of Washington Mikhail Sodin Tel Aviv University Anna Talitskaya Northwestern University Terence Tao University of California, Los Angeles Benjamin Weiss Hebrew University of Jerusalem Brian White Stanford University Mate Wierdl University of Memphis Xiangjin Xu University of Virginia Scott Zrebiec Johns Hopkins University Yunus E. Zeytuncu The Ohio State University Dapeng Zhan University of California, Berkeley Hua Zhang Northwestern University