topological dynamics and ergodic theory of automorphism...

Topological dynamics and ergodic theory ofautomorphism groups

Alexander S. Kechris

Harvard; November 18, 2013

Topological dynamics and ergodic theory of automorphism groups

Introduction

I will discuss some aspects of the topological dynamics and ergodictheory of automorphism groups of countable first-order structures andtheir connections with logic, finite combinatorics and probability theory.This is joint work with Omer Angel and Russell Lyons.

Fraısse theory

I will first review some basic concepts of Fraısse theory.

Definition

A class K of finite structures of the same signature is called a Fraısseclass if it satisfies the following properties:

(HP) Hereditary property.

(JEP) Joint embedding property.

(AP) Amalgamation property.

It is countable (up to ∼=).

It is unbounded.

Examples: finite graphs, finite linear orderings, f.d. vector spaces (over afinite field), finite Boolean algebras, finite rational metric spaces, finiteposets, ...

Fraısse theory

Definition

It is unbounded.

Fraısse theory

Definition

It is unbounded.

Fraısse theory

Joint embedding property (JEP)

Amalgamation property (AP)

Fraısse theory

Definition

A countable structure K is a Fraısse structure if it satisfies the followingproperties:

It is infinite.

It is locally finite.

It is ultrahomogeneous (i.e., an isomorphism between finitesubstructures can be extended to an automorphism of the wholestructure).

Examples: rational order, random graph, (countably) infinite dimensionalvector space (over a finite field), countable atomless Boolean algebra,rational Urysohn space.

Fraısse theory

Definition

A countable structure K is a Fraısse structure if it satisfies the followingproperties:

It is infinite.

It is locally finite.

It is ultrahomogeneous (i.e., an isomorphism between finitesubstructures can be extended to an automorphism of the wholestructure).

Examples: rational order, random graph, (countably) infinite dimensionalvector space (over a finite field), countable atomless Boolean algebra,rational Urysohn space.

Fraısse theory

Definition

For an infinite structure A, its age, denoted by Age(A), is the class offinite structures that can be embedded in A.

The age of a Fraısse structure is a Fraısse class and Fraısse showed thatconversely one can associate to each Fraısse class K a canonical Fraıssestructure K = Frlim(K), called its Fraısse limit, which is the uniqueFraısse structure whose age is equal to K. Therefore one has a canonicalone-to-one correspondence:

K 7→ Frlim(K)

between Fraısse classes and Fraısse structures whose inverse is:

K 7→ Age(K).

Fraısse theory

Definition

For an infinite structure A, its age, denoted by Age(A), is the class offinite structures that can be embedded in A.

The age of a Fraısse structure is a Fraısse class and Fraısse showed thatconversely one can associate to each Fraısse class K a canonical Fraıssestructure K = Frlim(K), called its Fraısse limit, which is the uniqueFraısse structure whose age is equal to K. Therefore one has a canonicalone-to-one correspondence:

K 7→ Frlim(K)

between Fraısse classes and Fraısse structures whose inverse is:

K 7→ Age(K).

Fraısse theory

Examples

finite graphs � random graph

finite linear orderings � 〈Q, <〉f.d. vector spaces � (countable) infinite-dimensional vector space(over a finite field)

finite Boolean algebras � countable atomless Boolean algebra

finite rational metric spaces � rational Urysohn space

Fraısse theory

Examples

Fraısse theory

Examples

Fraısse theory

Examples

Fraısse theory

Examples

Fraısse theory

Examples

Aut(A) as a topological group

For a countable structure A, we view Aut(A) as a topological groupwith the pointwise convergence topology. It is not hard to check thenthat it becomes a Polish group. In fact one can characterize these groupsas follows:

Theorem

For any Polish group G, the following are equivalent:

G is isomorphic to a closed subgroup of S∞, the permutation groupof N with the pointwise convergence topology.

G is non-Archimedean, i.e., admits a basis at the identity consistingof open subgroups.

G ∼= Aut(A), for a countable structure A.

G ∼= Aut(K), for a Fraısse structure K.

Aut(A) as a topological group

For a countable structure A, we view Aut(A) as a topological groupwith the pointwise convergence topology. It is not hard to check thenthat it becomes a Polish group. In fact one can characterize these groupsas follows:

Theorem

For any Polish group G, the following are equivalent:

G is isomorphic to a closed subgroup of S∞, the permutation groupof N with the pointwise convergence topology.

G is non-Archimedean, i.e., admits a basis at the identity consistingof open subgroups.

G ∼= Aut(A), for a countable structure A.

G ∼= Aut(K), for a Fraısse structure K.

Amenability of Aut(A)

We will now consider some aspects of the dynamics of automorphismgroups, especially the concept of amenability.

Definition

Let G be a topological group. A G-flow is a continuous action of G on acompact Hausdorff space. A group G is called amenable if every G-flowadmits an invariant (Borel probability) measure. It is called extremelyamenable if every G-flow admits an invariant point.

Remark

No non-trivial locally compact group can be extremely amenable.

Definition

Remark

Definition

Remark

Extreme amenability and Ramsey theory

In a paper of K-Pestov-Todorcevic (2005) a duality theory was developedthat relates the Ramsey theory of Fraısse classes (sometimes calledstructural Ramsey theory) to the topological dynamics of theautomorphism groups of their Fraısse limits.

Structural Ramsey theory is a vast generalization of the classical Ramseytheorem to classes of finite structures. It was developed primarily in the1970’s by: Graham, Leeb, Rothschild, Nesetril-Rodl, Promel, Voigt,Abramson-Harrington, ...

In a paper of K-Pestov-Todorcevic (2005) a duality theory was developedthat relates the Ramsey theory of Fraısse classes (sometimes calledstructural Ramsey theory) to the topological dynamics of theautomorphism groups of their Fraısse limits.

Structural Ramsey theory is a vast generalization of the classical Ramseytheorem to classes of finite structures. It was developed primarily in the1970’s by: Graham, Leeb, Rothschild, Nesetril-Rodl, Promel, Voigt,Abramson-Harrington, ...

Definition

A class K of finite structures (in the same signature) has the Ramseyproperty (RP) if for any A ≤ B in K, and any n ≥ 1, there is C ≥ B inK, such that

C → (B)An .

Examples of classes with Ramsey property:

finite linear orderings (Ramsey)

finite Boolean algebras (Graham-Rothschild)

finite-dimensional vector spaces over a given finite field(Graham-Leeb-Rothschild)

finite ordered graphs (Nesetril-Rodl)

finite ordered rational metric spaces (Nesetril)

Definition

C → (B)An .

Definition

C → (B)An .

Definition

C → (B)An .

Definition

C → (B)An .

Definition

C → (B)An .

Definition

C → (B)An .

One of the consequences of the duality theory is the followingcharacterization of extreme amenability of automorphism groups.

Theorem (KPT)

The extremely amenable automorphism groups are exactly theautomorphism groups of ordered Fraısse structures whose age satisfiesthe Ramsey Property.

Examples

The automorphism groups of the following structures are extremelyamenable:

random ordered graph

(Pestov) rational order

lex. ordered infinite-dimensional vector space (over a finite field)

lex. ordered countable atomless Boolean algebra

rational ordered Urysohn space

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Theorem (KPT)

Examples

Hrushovski structures

Clearly every extremely amenable group is amenable. There are howevermany amenable automorphism groups that are not extremely amenable.Many such examples arise in the context of the Hrushovski Property.

Definition

Let K be a Fraısse class of finite structures. We say that K is aHrushovski class if for any A in K there is B in K containing A suchthat any partial automorphism of A extends to an automorphism of B.

Some basic examples of such classes are the pure sets, graphs(Hrushovski), hypergraphs and Kn-free graphs (Herwig), rational valuedmetric spaces (Solecki), finite dimensional vector spaces over finite fields,etc.

Definition

Let K be a Fraısse class of finite structures and K its Fraısse limit. If Kis a Hrushovski class, then we say that K is a Hrushovski structure.

Definition

This turns out to be a property of automorphism groups:

Proposition

Let K be a Fraısse class of finite structures and K its Fraısse limit.Then the following are equivalent

K is a Hrushovski structure.

Aut(K) is compactly approximable, i.e., there is a increasingsequence Kn of compact subgroups whose union is dense in theautomorphism group.

In particular the automorphism group of a Hrushovski structure isamenable. Thus S∞ and the automorphism groups of the random graph,random n-uniform hypergraph, random Kn-free graph, rational Urysohnspace, (countably) infinite-dimensional vector space over a finite field,etc., are amenable (but not extremely amenable).

Proposition

Non-amenable groups

At the other end of the spectrum there are also automorphism groupsthat are not amenable. These include the following:

Theorem (K-Sokic)

The automorphism groups of the random poset and random distributivelattice are not amenable.

Theorem (Malicki)

The automorphism group of the random lattice is not amenable.

Non-amenable groups

Theorem (K-Sokic)

Theorem (Malicki)

Non-amenable groups

Theorem (K-Sokic)

Theorem (Malicki)

Unique ergodicity

I am interested here in the ergodic theory of flows of automorphismgroups and especially in the phenomenon of unique ergodicity.Let G be a topological group and X a G-flow. Consider G-invariant(Borel probability) measures in such a flow.

Definition

A G-flow is uniquely ergodic if it admits a unique invariant measure(which must then be ergodic).

Unique ergodicity

I am interested here in the ergodic theory of flows of automorphismgroups and especially in the phenomenon of unique ergodicity.Let G be a topological group and X a G-flow. Consider G-invariant(Borel probability) measures in such a flow.

Definition

A G-flow is uniquely ergodic if it admits a unique invariant measure(which must then be ergodic).

Unique ergodicity

Recall that a flow is called minimal if every orbit is dense or equivalentlyif is has no proper subflows. Every flow contains a minimal subflow.

Definition

Let G be a topological group. We call G uniquely ergodic if everyminimal flow admits a unique invariant measure (which must then beergodic).

Remark: The assumption of minimality is necessary because in general aflow has many minimal subflows which are of course pairwise disjoint.Note also that every uniquely ergodic group is amenable.

Clearly every extremely amenable Polish group is uniquely ergodic and sois every compact Polish group. On the other hand Benjamin Weiss hasshown that no infinite countable (discrete) group can be uniquely ergodicand he believes that this extends to Polish locally compact, non-compactgroups although this has not been verified in detail.

Unique ergodicity

Definition

Unique ergodicity

Definition

Unique ergodicity

Definition

Universal minimal flows

In order to understand better the concept of unique ergodicity we need todiscuss first the idea of a universal minimal flow.

A homomorphism between two G-flows X,Y is a continuous G-mapπ : X → Y . If Y is minimal, then π must be onto. An isomorphism is abijective homomorphism.

Theorem

For any G, there is a minimal G-flow, M(G), called its universal minimalflow with the following property: For any minimal G-flow X, there is ahomomorphism π : M(G)→ X. Moreover M(G) is uniquely determinedup to isomorphism by this property.

Theorem

The following is a consequence of the Hahn-Banach Theorem.

Proposition

Let G be an amenable group. Then G is uniquely ergodic iff M(G) isuniquely ergodic.

So it is enough to concentrate on the universal minimal flow.

Proposition

If G is compact, then M(G) = G. If G is non-compact but locallycompact, then M(G) is extremely complicated, e.g., it is non-metrizable.However, by definition G is extremely amenable iff M(G) trivializes!

This leads to a general problem in topological dynamics:

For a given G, can one explicitly determine M(G) and show that it ismetrizable?

Universal minimal flows of automorphism groups

The duality theory of K-Pestov-Todorcevic provides tools for computingthe universal minimal flows of automorphism groups of Fraısse structures.We will discuss this next.

Order expansions of Fraısse classes

Consider a Fraısse class K. A Fraısse class K∗ is an order expansion of Kif K∗ consists of structures of the form 〈A, <〉, where A ∈ K and < is alinear ordering on (the universe of) A. In this case, if 〈A, <〉 ∈ K∗ wecall < a K∗-admissible ordering on A. The order expansion K∗ of K isreasonable if for every A,B ∈ K, with A ⊆ B and any K∗-admissibleordering < on A, there is a K∗-admissible ordering <′ on B such that<⊆<′.

If K is a Fraısse class with K = Flim(K) and K∗ is a reasonable, orderexpansion of K, we denote by XK∗ the space of linear orderings < on Ksuch that for any finite substructure A of K, < |A is K∗-admissible onA. We call these the K∗-admissible orderings on K. They form acompact, metrizable, non-empty subspace of 2K2

(with the producttopology) on which the group G = Aut(K) acts continuously, thus XK∗is a G-flow.

Examples

K = finite graphs, K = R; K∗ = finite ordered graphs. Then XK∗is the space of all linear orderings of the random graph.

K = finite sets, K = 〈N〉; K∗ = finite orderings. Then XK∗ is thespace of all linear orderings on N .

K = f.d. vector spaces over a fixed finite field, K = V ∞; K∗ = lex.ordered f.d. vector spaces. Then XK∗ is the space of all “lex.orderings” on V ∞.

K = finite posets, K = P ; K∗ = finite posets with linear extensions.Then XK∗ is the space of all linear extensions of the random poset.

Examples

Beyond the Ramsey Property, there is an additional property of classes offinite structures that was introduced by Nesetril and Rodl in the 1970’sand played an important role in the structural Ramsey theory.

Definition

If K∗ is an order expansion of K, we say that K∗ satisfies the orderingproperty (OP) if for every A ∈ K, there is B ∈ K such that for everyK∗-admissible orderings < on A and <′ on B, 〈A, <〉 can be embeddedin 〈B, <′〉.

In all the examples of the previous page we have the ordering property.

Definition

Calculation of universal minimal flows

Theorem (KPT)

Let K be a Fraısse class and K∗ a reasonable order expansion of K. Thenif G is the automorphism group of the Fraısse limit of K the following areequivalent:

XK∗ is the universal minimal flow of the automorphism group of G.

K∗ has the Ramsey Property and the Ordering Property.

Examples

K = finite graphs, K = R; K∗ = finite ordered graphs. Then thespace of all linear orderings of the random graph is the UMF of itsautomorphism group.

K = finite sets, K = 〈N〉; K∗ = finite orderings. Then the space ofall linear orderings on N is the UMF of S∞(Glasner-Weiss).

K = f.d. vector spaces over a fixed finite field, K = V ∞; K∗ = lex.ordered f.d. vector spaces. Then the space of all “lex. orderings” onV ∞ is the UMF of its general linear group.

K = finite posets, K = P ; K∗ = finite posets with linearextensions. Then the space of all linear extensions of the randomposet is the UMF of its automorphism group.

Examples

Unique ergodicity revisited

Let K be a Fraısse class and K∗ a reasonable order expansion of K thathas the Ramsey Property and the Ordering Property. We will say thenthat K∗ is a companion of K. It was shown in the paper of KPT thatsuch a companion, when it exists, is essentially unique.

Thus we have seen that when K has a companion class K∗, and thishappens for many important examples, then the UMF of theautomorphism group G of its Fraısse limit is the compact, metrizablespace XK∗ . Thus the unique ergodicity of G is equivalent to the uniqueergodicity of XK∗ . This can then be seen to be equivalent to thefollowing probabilistic notion.

Let K be a Fraısse class and K∗ a reasonable order expansion of K thathas the Ramsey Property and the Ordering Property. We will say thenthat K∗ is a companion of K. It was shown in the paper of KPT thatsuch a companion, when it exists, is essentially unique.

Thus we have seen that when K has a companion class K∗, and thishappens for many important examples, then the UMF of theautomorphism group G of its Fraısse limit is the compact, metrizablespace XK∗ . Thus the unique ergodicity of G is equivalent to the uniqueergodicity of XK∗ . This can then be seen to be equivalent to thefollowing probabilistic notion.

Definition

Let K∗ be a companion of K. A random, consistent K∗-admissibleordering is a map that assigns to each structure A ∈ K a probabilitymeasure µA on the (finite) space of K∗-admissible orderings on A, whichis isomorphism invariant and has the property that if A ⊆ B, then µB

projects by the restriction map to µA.

We now have:

Proposition (AKL)

Let K∗ be a companion of K. Then amenability of the automorphismgroup G of the Fraısse limit of K is equivalent to the existence of arandom, consistent K∗-admissible ordering and unique ergodicity of G isequivalent to the uniqueness of a random, consistent K∗-admissibleordering.

Example: graphs

Definition

We now have:

Proposition (AKL)

Example: graphs

Definition

We now have:

Proposition (AKL)

Example: graphs

Unique ergodicity as a quantitative version of the OrderingProperty

Interestingly it turns out that unique ergodicity fits well in the frameworkof the duality theory of KPT (which originally was developed in thecontext of topological dynamics). In many cases it can simply be viewedas a quantitative version of the Ordering Property.

Definition (AKL)

Let K∗ be a companion of K. We say that K∗ satisfies the QuantitativeOrdering Property (QOP) if the following holds:

There is an isomorphism invariant map that assigns to each structureA∗ = 〈A, <〉 ∈ K∗ a real number ρ(A∗) in (0, 1] such that for everyA ∈ K and each ε > 0, there is a B ∈ K and a nonempty set ofembeddings E(A,B) of A into B with the property that for eachK∗-admissible ordering < of A and each K∗-admissible ordering <′ of Bthe proportion of embeddings in E(A,B) that preserve <,<′ is equal toρ(〈A, <〉), within ε.

Interestingly it turns out that unique ergodicity fits well in the frameworkof the duality theory of KPT (which originally was developed in thecontext of topological dynamics). In many cases it can simply be viewedas a quantitative version of the Ordering Property.

Definition (AKL)

Let K∗ be a companion of K. We say that K∗ satisfies the QuantitativeOrdering Property (QOP) if the following holds:

There is an isomorphism invariant map that assigns to each structureA∗ = 〈A, <〉 ∈ K∗ a real number ρ(A∗) in (0, 1] such that for everyA ∈ K and each ε > 0, there is a B ∈ K and a nonempty set ofembeddings E(A,B) of A into B with the property that for eachK∗-admissible ordering < of A and each K∗-admissible ordering <′ of Bthe proportion of embeddings in E(A,B) that preserve <,<′ is equal toρ(〈A, <〉), within ε.

Theorem (AKL)

Let K∗ be a companion of K, let G be the automorphism group of theFraısse limit of K and assume that G is amenable. Then QOP impliesthe unique ergodicity of G. Moreover, if K is a Hrushovski class, QOP isequivalent to the unique ergodicity of G.

Theorem (AKL)

The QOP holds for the following Fraısse classes :

ordered graphs

ordered Kn-free graphs

ordered n-uniform hypergraphs

rational ordered metric spaces

In particular, in all these cases there is a unique random, consistentordering, namely the uniform one.

The proofs use probabilistic arguments (deviation or concentrationinequalities).

Theorem (AKL)

ordered graphs

Theorem (AKL)

ordered graphs

For example, if K is the class of finite graphs, we establish QOP byshowing that for any finite graph A with n vertices and ε > 0, there is agraph B, containing a copy of A, such that given any orderings < on Aand <′ on B, the proportion of all embeddings of A into B thatpreserve the orderings <,<′ is, up to ε, equal to 1/n!.

Proving unique ergodicity

Theorem (AKL, except for S∞)

The following automorphism groups are uniquely ergodic:

S∞ (Glasner-Weiss)

The isometry group of the Baire space

The general linear group of the (countably) infinite-dimensionalvector space over a finite field

The automorphism group of the random graph

The automorphism group of the random Kn-free graph

The automorphism group of the random n-uniform hypergraph

The isometry group of the rational Urysohn space

Proving unique ergodicity

Theorem (AKL, except for S∞)

The following automorphism groups are uniquely ergodic:

S∞ (Glasner-Weiss)

The isometry group of the Baire space

The general linear group of the (countably) infinite-dimensionalvector space over a finite field

The automorphism group of the random graph

The automorphism group of the random Kn-free graph

The automorphism group of the random n-uniform hypergraph

The isometry group of the rational Urysohn space

Unique Ergodicity Problem

In fact I do not know any counterexample to the following problem:

Problem (Unique Ergodicity Problem)

Let G be an amenable automorphism group of a countable structure witha metrizable universal minimal flow. Is G uniquely ergodic?

Next I will consider the problem of determining the support of the uniquemeasure (in the uniquely ergodic case).

Generic Orbit Problem

Definition

Let X be a G-flow. A comeager orbit of this action is called a genericorbit. (It is of course unique if it exists.) We say that G has the genericorbit property if every minimal G-flow has a generic orbit.

It turns out that G has the generic orbit property iff its universal minimalflow has a generic orbit. Using this one can show:

Theorem (AKL)

Let K be a Fraısse class that admits a companion K∗. Then theautomorphism group of the Fraısse limit of K has the generic orbitproperty.

Remark

Again it can be shown that no non-compact locally compact Polish groupcan satisfy the generic orbit property.

Definition

Theorem (AKL)

Remark

Definition

Theorem (AKL)

Remark

Definition

Theorem (AKL)

Remark

I do not know any counterexample to the following problem:

Problem (Generic Orbit problem)

Let G be an automorphism group of a countable structure with ametrizable universal minimal flow. Does G have the generic orbitproperty?

I do not know any counterexample to the following problem:

Problem (Generic Orbit problem)

Let G be an automorphism group of a countable structure with ametrizable universal minimal flow. Does G have the generic orbitproperty?

Unique Ergodicity - Generic Orbit Problem

Merging the two previous problems, we can ask whether an even strongerproperty is true, which specifies the support of the unique invariantmeasure.

Problem (Unique Ergodicity - Generic Orbit Problem)

Let G be an amenable automorphism group of a countable structure witha metrizable universal minimal flow. Is G uniquely ergodic, has thegeneric orbit property and moreover in every minimal G-flow the uniqueinvariant measure is supported by the generic orbit orbit?

A positive answer has been obtained in many cases, e.g., S∞(Glasner-Weiss); the automorphism group of the random graph, randomn-uniform hypergraph, random Kn-free graph, rational Urysohn space,etc. (AKL). Note that when Unique Ergodicity – Generic Orbit holds onehas the interesting phenomenon that measure and category agree insteadof being, as usual, orthogonal.

Unique Ergodicity - Generic Point Problem

I will finish with two very recent interesting (and contrasting) answers tothe Unique Ergodicity-Generic Orbit Problem.

Andrew Zucker (Caltech undergraduate) has surprisingly shown that forthe general linear group of the (countably) infinite-dimensional vectorspace over a finite field, which is uniquely ergodic and has the genericorbit property, the unique invariant measure of its universal minimal flowdoes not live on the generic orbit! Thus the Unique Ergodicity - GenericOrbit Problem has a negative answer in general.

On the other hand, Andras Pongracz subsequently showed that when aFraısse class K admits an order-forgetful companion K∗ and theautomorphism group of its Fraısse limit is amenable, then it is uniquelyergodic, has the generic orbit property and the unique invariant measurelives on the generic orbit, provided that the language of K is relational.Thus the Unique Ergodicity - Generic Orbit Problem has a positiveanswer in this situation. Vector spaces satisfy all these properties but thelanguage is not relational!

topological dynamics and ergodic theory of automorphism...

Documents