applications of descriptive set theory to functional analysis and

IntroductionApplication to functional analysis

sketch of proofApplication to Dynamics

Applications of Descriptive Set Theory toFunctional Analysis and Dynamics

S. Jackson

Department of MathematicsUniversity of North Texas

August, 2013SUMIRFAS, College Station

S. Jackson Applications of Descriptive Set Theory to Functional Analysis and Dynamics



Overview

We discuss to applications of descriptive set theory to classifyingcomplexity, one in functional analysis, and one in dynamics.

The first result concerns the complexity of the uniformhomeomorphism relation between separable Banach spaces. It isjoint with Su Gao and Bunyamin Sari.

The second result concerns the complexity of a natural idealassociated to attractors from iterated function systems.




Basic objects of study in analysis are the Polish spaces X , thecomplete separable metric spaces.

Recall (X ,B) is a standard Borel space if B is a σ-algebra on Xsuch that (X ,B) is isomorphic to the algebra of Borel sets in somePolish space.

Any Borel subset of a Polish space is a standard Borel space.

Examples: Rnstd, ωω, 2ω, probability measures on a Polish space,

separable Banach spaces, F (X ).




Any two uncountable Polish spaces are Borel isomorphic, so amongthe standard Borel spaces the only invariants are the cardinalities1, 2, . . . , ω, c .

In Polish spaces, one classifies the complexities of sets through thenotion of a reduction.

Let X , Y be Polish spaces and A ⊆ X , B ⊆ Y . We say A isWadge reducible to B, A ≤w B if there is a continuous functionf : X → Y such that A = f −1(B), that is,

x ∈ A⇔ f (x) ∈ B

[ This definition is reasonable if X , Y are 0-dimensional, otherwisecan use coarser notion of Borel reduction].




For sets in ωω complexity classes fall into a wellorderded hierarchyvia Wadge’s lemma and the Martin-Monk theorems.

Wadge’s Lemma: Assume sufficient determinacy. Then forA,B ⊆ ωω either A ≤w B or B ≤w A.

So, if we identify the degree of A with Ac , then the degrees arelinearly orderded.

Martin-Monk: Assume sufficient determinacy. Then the Wadgedegrees are wellorderded.

So, the Wadge hierarcy gives a complete classification of thecomplexity degrees for sets in Polish space.




We wish a notion of complexity to classify objects more generalthan subsets of a Polish space.

Let E be an equivalence relation on a Polish space X . Thecorresponding quotient space is X/E .

Objects of the form X/E represent all objects for which there is asurjection of R onto the object. We usually restrict to Borelequivalence relations E (though the results usually apply toarbitrary E given determinacy).

In general, there is no reasonable (definable) way to regard X/E asa subset of a standard Borel space.

We generalize the notion of reduction to introduce notion ofcomplexity for equivalence relations on standard Borel spaces.




Let E , F be Borel equivalence relations on standard Borel spacesX , Y .

Definition

I We say E is reducible to F , E ≤ F , if there is a Borel functionf : X → Y such that xE y ⇔ f (x)F f (y).

I We say E is bi-reducible with F , E ∼ F , if E ≤ F and F ≤ E .

I We say E is emdeddable into F , E v F , if in addition f isone-to-one.

Note that a reduction gives a definable injection from X/E toY /F so reduction can be viewed as a notion of definablecardinality for these quotient spaces.




We say E is a countable (Borel) equivalence relation if all classesof E are countable.

If G is a Polish group and G acts on X , then the orbit equivalencerelation EG is defined by

xEG y ⇔ ∃g ∈ G (g · x = y).

The Feldman-Moore theorem says that every countable Borelequivalence relation is given by the Borel action of a countablegroup G . The case G = Z is the classical case of discrete-timedynamics.

So, we can study the equivalence relations EG group by group.




The simplest equivalence relations are the smooth or tame ones.

DefinitionE is smooth if there is a Borel reduction of E to equality relationon a Polish space.

So, for a smooth E , X/E can be regarded as a subset of astandard Borel space.

For countable Borel E , smooth is the same as saying there is aBorel selector for E .




DefinitionE0 is the equivalence relation on 2ω given by

xE0 y ⇔ ∃n ∀m ≥ n (x(m) = y(m)).

The Harrington-Kechris-Louveau theorem says that if E is a Borelequivalence relation then either E is smooth or E0 v E .

So, there is no complexity class of equivalence relation strictlybetween the smooth relation E= and E0.




If G is a Polish group, G acts of F (G ) by the shift action

g · F = {gf : f ∈ F}

We can view this action as being on 2G by

g · x(h) = x(g−1h)

We call this the Bernoulli (left) shift action of G on 2G . When Gis countable, 2G is a compact Polish space in the natural producttopology.




The Bernoulli shift is more or less universal for

FactLet E the the orbit equivalence relation for a Borel action of thecountable group G on a Polish space X . Then

E ≤ E ((2ω)G ) ≤ E (2G×Z).

A theorem of Becker-Kechris says that for general Polish G theω-product of the shift action of G on F (G ) is universal for BorelG -actions.

There is also a universal Polish group (e.g., ISO(U)), so that thereis a universal Polish group action.




Countable Equivalence Relations

Theorem (Dougherty-J-Kechris)

The shift action of F2 on 2F2 is a universal countable Borelequivalence relation, that is, E ≤ EF2 for any countable Borel E .

DefinitionA countable Borel equivalence relation E is hyperfinite if E is theincreasing union of relations En with finite classes.

Theorem (Slaman-Steel)

The following are equivalent:

I E is hyperfinite.

I E = EG where G = Z.

I The classes of E can be uniformly Borel ordered in type Z (orare finite).




Theorem (Weiss)

Every Borel action by Zn is hyperfinite.

Theorem (Gao-J)

Every Borel action by a countable abelian group is hyperfinite.

Hyperfiniteness problem (Kechris): Is every equivalence relationgenerated by a Borel action of an amenable group hyperfinite?

Adams-Kechris showed that the structure of the countable Borelequivalence relations above E0 is complicated.




Some Non-countable Relations

I Eω0 : Does not embed into any Σ0

3 equivalence relation.

I `p, 1 ≤ p ≤ ∞. Given by action of Polish group. ByDougherty-Hjorth `p ≤ `q for p ≤ q.

I `∞ by Rosendal is universal for Kσ equivalence relations.

I E1: Defined as E0 except on R.

I =+: Equality on countable sets of reals. Embeds anycountable equivalence relations, given by S∞ action.




The collection B of separable Banach spaces can be viewed as aPolish space by identifying separable Banach spaces with closedsubspaces of C [0, 1]. The subset of F (C [0, 1]) corresponding tolinear subspaces is a Gδ.

The collection Bb of separable Banach spaces with basis (X ,B)can also be viewed as a Polish space.

I One way is to use a universal basis, so we identify (X ,B) withan x ∈ 2ω.

I Or can identify (X , (bi )) with the sequence ‖∑k

i=1 ani bi‖X ,where (an1, . . . , a

nk) enumerates Q<ω. This identifies Bb with a

closed subset of Rω.

There are Borel functions taking codes for members of Bb in onecoding to codes in the other coding.




Some known classifications

(Ferenczi, Louveau, Rosendal) Isomorphism on separable Banachspaces is bireducible with Σ1

1-complete equivalence relation.

(Rosendal) Equivalence of bases is complete Kσ equivalencerelation (`∞).

(Melleray) Isometry between separable Banach spaces is universalorbit equivalence relation.

(Gao-J-Sari) Uniform homeomorphism between separable Banachspaces is at least `∞.




id(2ω)qqE0

qEG , G amenable

hyperfinite?

'

&

$

%

qqE∞

qid(2ω)

qE0

QQ

QQQq

Eω0

AAAq

E∞

��q̀1

��

qE1

q=+

JJJJJ

q`∞(equivalence

of bases)

��

��

JJJJJJJ

qEΣ11 (isomorphism)

JJJJJJJqE∞G (isometry)




Recall that for linearly isomorphic Banach spaces X , Y , theBanch-Mazur distance is defined by

d(X ,Y ) = inf{‖T‖‖T−1‖ : T : X → Y is an isomorphism }

DefinitionX , Y are locally equivalent if there is a C > 0 such that for everyfinite dimensional subspace E of X there is a subspace F of Ywith d(E ,F ) ≤ C and vice-versa. We write X ≡L Y .

Theorem (Ribe)

If X , Y are uniformly homeomorphic then X ≡L Y .




FactLocal equivalence is a Σ3

0 equivalence relation on B or on Bb.

Corollary

Eω0 , and hence =+ is not reducible to ≡L on either B or Bb.

In fact we have:

Theorem (Gao-J-Sari)

≡L (on either B or Bb) is bi-reducible with `∞.

On the other hand, local equivalence differs from uniformhomeomorphism:

Theorem (GJS)

There is a single Borel class C of separable Banach spaces underuniform homeomorphism such that =+ is reducible to theisomorphism relation on C.




Sketch of proof of reducibility

Let Ii = [li , ri ] ⊆ (1, 2) with |Ii | = 12i+1 , and let ni ∈ Z+.

Let ~p = (pi ) with pi ∈ Ii .

Let

S~p =

( ∞∑i=1

`nipi

)2

Fix a sequence bi ∈ Z+ with lim sup bi =∞.

Let σi : [0, bi ]→ Ii be an affine bijection.




1 2

Ii

li ri

0 bi

σi

· · ·

Figure: Basic setup for the embedding

I Ii = [li , ri ], |Ii | = 2−(i+1)

I σi : [0, bi ]→ Ii




We choose the ni carefully, they must grow fast, but not too fast.

Letni = 2bi2

i

We can easily reduce `∞(R) to `∞([0, bi ]), so we use the latter.

Given x ∈∏

[0, bi ], we define π(x) to be the separable Banachspace π(x) = S~p,~n where ~p ∈

∏Ii is given by

pi = σi (x(i))

We check that π is a reduction from `∞([0, bi ]) to the uniformhomeomorphism relation on B.




Note that (we assume the bi are increasing)

n1/ri+1

i+1 ≥ n1/2i+1 = 2bi+12

i ≥ ni ≥ n1/lii

Lemma

S~p, S~q are uniformly homeomorphic iff supi n| 1pi− 1

qi|

i <∞.

Proof. If n| 1pi− 1

qi|

i < C for all i , then d(`nipi , `niqi

) ≤ C , just take theidentity map.

Let p = pi0 , q = qi0 , n = ni0 , and assume p < q.




Let T : `np → S~q be a linear embedding. We show that

‖T‖‖T−1‖ ≥ C (n1/p−1/q) for some absolute constant C .

To do this we consider the type 2, n constants of `np and S~q.

Recall the type p, n constant Tp,n(X ) of a Banach space X is thesmallest constant such that for all n vectors x1, . . . , xn ∈ X wehave: (

Aveεi=±1

‖n∑

i=1

εixi‖2)1/2

≤ Tp,n(X )

(n∑

i=1

‖xi‖p)1/p

X is of type p if Tp(X ) = supn Tp,n(X ) <∞.




If T : X → Y is a linear embedding then

Tp,n(X ) ≤ ‖T‖‖T−1‖Tp,n(Y ).

We have (c.f. Tomczak-Jaegermann):

I Tp(`nq) ≥ nmax(0,1/q−1/p)

I Tp(`nq) ≤ cq1/2nmax(0,1/q−1/p) (1 ≤ q <∞)

I For n ≤ k, Tp,n(`kq) ≤ cq1/2nmax(0,1/q−1/p)




Fix T : `np → S~q. We estimate T2,n(`np).

On the one hand it is at least n1/p−1/2.

On the other hand, it is ≤ ‖T‖‖T−1‖T2,n(S~q).

A computation using the previous facts and n2/qii < n2/q for i < i0

gives that

T2,n(S~q) ≤√

2cn1/q−1/p

and so ‖T‖‖T−1‖ ≥ 1√2c

n1/p−1/q.

This proves the lemma.




This shows `∞ = `∞(R) reduces to uniform homeomorphism onseparable Banach spaces.

We also get that `∞ is bi-reducible with ≡L.

For this we need the reverse reduction from ≡L to `∞.

This follows from the following fact.

For any Polish (X , d) let ~xFX ~y iff the d-Hausdorff distancebetween {x(i)} and {y(i)} is finite.

FactFor any Polish (X , d), FX ≤ `∞.




Sketch of proof of lower bound

Dynamics

Let fi : Rn → Rn, i = 1, . . . , k, be contraction mappings, that is‖fi (x)− fi (y)‖ ≤ c‖x − y‖ where c < 1.

There is a unique compact set K = K (f1, . . . , fn) such thatK = f1(K ) ∪ · · · ∪ fk(K ).

We say ~f satisfies the porosity condition if there is a p > 0 suchthat for all balls B of radius r there is a sub-ball B ′ ⊆ B of radiuspr such that B ′ ∩ K = ∅.

If we fix a ball B(r) and a c < 1, then the set of ~f which arecontractions of B(r) with constant c and is a Polish space F (r , c),and the set of ~f satisfying the porosity condition is a closed subsetof F (r , c).





We let IFS denote the K ∈ F (Rn) of the form K = K (~f ) for someiterated function system ~f satisfying the porosity condition.

We let Iifs denote the σ-ideal generated by the sets in IFS . So,Iifs ⊆ X is a σ-ideal of closed subsets of Rn.

We recall the following dichotomy theorem of Kechris and Woodin:

Theorem (Kechris,Woodin)

Let I be a Π11 σ-ideal of closed sets in a compact metric space.

Then either I is Π02 or I is Π1

1-complete.





Every F ∈ Iifs is meager. However the meager ideal is Π02. A

straightforward computation shows that Iifs is Σ12.

TheoremIifs is Π1

1-complete.

More generally, Let P = {pα}α<λ be a family of porosityfunctions, pα : ω → ω.

Let IPifs be the σ-ideal generated by the K (~f ) which have a porosityfunction dominated by a pα ∈ P.





TheoremLet P be be given and assume there is a g : ω → ω such that

∀p ∈ P ∃∞n (p(n + 1)

p(n)< g(n)).

Then IPifs is Π11-hard.

Corollary

There is an uncountable Borel P such that IPifs is Π11-hard.

When P is countable we get the upper bound: IPifs is Π11.





So we have:

I If |P| < b, then IPifs is Π11 hard.

I If P is countable then IPifs is Π11-complete.

I If P = ωω then IPifs is Π02.

I There are uncountable Borel P for which IPifs is Π11 hard.

Corollary

The ideal ICifs of all K (~f ) which have a computable porosityfunction is Π1

1-complete.





Some questions

QuestionAre there any ∆1

1 P for which IPifs is Σ12-complete?

QuestionFor which ∆1

1 P is IPifs Π02, Π1

1, Σ12-complete?





Enough to consider n = 1, in fact work in [0, 1].

Fix a reasonable homeomorphism h between ωω and Irr ⊆ [0, 1].

Let Tr be the Polish space of trees on ω and W ⊆ Tr the set ofwellfounded trees. So, W is Π1

1-complete.

We define a continuous map π : T → K ([0, 1]) reducing W to Iifs.

If T ∈W then π(T ) will be countable, and hence in Iifs. If T /∈Wthen we will have π(T ) ⊆ π0(T ) ∪Q where π0(T ) ⊆ Irr ∩ [0, 1] isa closed subset of Irr.





We actually define π0 : Tr → Tr .

Fix a function b : Q× ω<ω → ω such that for rational porositynumber p and s ∈ ω<ω, we have that for any iterated functionsystem ~f with porosity ≥ p there is a sequence t ∈ ω<ω of length≤ b(p, s) such that h(sat) ∩ K (~f ) = ∅.

This uses the fact that h is reasonable. Actually, b(p, s) dependsonly on p and |s|.

For each n fix a continuous surjection h1/n from ωω to the Polish

space of iterated function systems on [0, 1] with porosity p ≥ 1n

and c ≤ 1− 1n .





0 0

s0

t0v0•

p0

s1

t1v1•

p1

I si build a branch through T .

I ti build an IFS . Let t̄ = t̄0at̄1

a · · · (drop first digits) and lett ′i = (t̄)i = t̄〈i ,0〉

at̄〈i ,1〉 · · · .I vi get out of K (~f ).

I h1/ti (0)(t ′i ) is consistent with being an iterated function system~f such that K (~f ) ∩ Nu0av0a···avi = ∅.

I |vi+1| = b(p, u0av0

a · · ·aui+1) where p = 1ti+1(0)

.


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