Borel Automorphisms Research Articles

Coboundaries of commuting Borel automorphisms

We show that if \(S\), \(T\) are two commuting automorphisms of a standard Borel space such that they generate a free Borel \(\mathbb{Z}^2\)-action then \(S\) and \(T\) do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel \(\mathbb{Z}^d\)-actions.

On a Universal Borel Adic Space

We prove that the so-called uniadic graph and its adic automorphism are Borel universal, i.e., every aperiodic Borel automorphism is isomorphic to the restriction of this automorphism to a subset invariant under the adic transformation, the isomorphism being defined on a universal (with respect to the measure) set. We develop the concept of basic filtrations and combinatorial definiteness of automorphisms suggested in our previous paper.

Every Borel automorphism without finite invariant measures admits a two-set generator

We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator. This implies that if the entropies of invariant probability measures of a Borel system are all less than log k , then the system admits a k -set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.

Baire measurable paradoxical decompositions via matchings

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in R3 using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann–Day problem: if a is a nonamenable action of a group on a Polish space X by Borel automorphisms, then there is a free Baire measurable action of F2 on X which is Lipschitz with respect to a.

Descriptive Kakutani equivalence

We consider a descriptive set-theoretic analog of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two aperiodic Borel automorphisms of uncountable Polish spaces. Using this, we classify all Borel ℝ -flows up to C^∞ -time-change isomorphism. We then extend the notion of descriptive Kakutani equivalence to all (not necessarily injective) Borel functions, and provide a variety of results leading towards a complete classification. The main technical tools are a series of Glimm–Effros and Dougherty–Jackson–Kechris-style embedding theorems.

Classifying Borel automorphisms

§1. Introduction. This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence:Definition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g, if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ.We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2ω (with the product topology) as our representative space.

Exchangeable measures for subshifts

Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when Ω = S N . We apply the ergodic theory of equivalence relations to study the case Ω ≠ S N , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [ 0 , 1 ] (a non-Markovian constraint).

On the non-existence of certain group topologies

Minimal Hausdorff (Baire) group topologies of certain groups of transformations naturally occurring in analysis are studied. The results obtained are subsequently applied to show that, e.g., the homeomorphism groups of the rational and of the irrational numbers carry no Polish group topology. In answer to a question of A.S. Kechris it is shown that that the group of Borel automorphisms of R cannot be a Polish group either.

Topologies on the group of Borel automorphisms of a standard Borel space

The paper is devoted to the study of topologies on the group $\text{\rm Aut}(X,{\Cal B})$ of all Borel automorphisms of a standard Borel space $(X, {\mathcal B})$. Several topologies are introduced and all possible relations between them are found. One of these topologies, $\tau$, is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of $\text{\rm Aut}(X,{\mathcal B})$ and find their closures. In particular, we describe closures of subsets formed by odometers, periodic, aperiodic, incompressible, and smooth automorphisms with respect to the defined topologies. It is proved that the set of periodic Borel automorphisms is dense in $\text{\rm Aut}(X,{\mathcal B})$ (Rokhlin lemma) with respect to $\tau$. It is shown that the $\tau$-closure of odometers (and of rank $1$ Borel automorphisms) coincides with the set of all aperiodic automorphisms. For every aperiodic automorphism $T\in \text{\rm Aut}(X,{\mathcal B})$, the concept of a Borel-Bratteli diagram is defined and studied. It is proved that every aperiodic Borel automorphism $T$ is isomorphic to the Vershik transformation acting on the space of infinite paths of an ordered Borel-Bratteli diagram. Several applications of this result are given.

Smooth automorphisms and path-connectedness in Borel dynamics

Let Aut(X, B) be the group of all Borel automorphisms of a standard Borel space ( X, B). We study topological properties of Aut(X, B) with respect to the uniform and weak topologies, τ and p, defined in [Bezuglyi S., Dooley A.H., Kwiatkowski J., Topologies on the group of Borel automorphisms of a standard Borel space, Preprint 2003]. It is proved that the class of smooth automorphisms is dense in ( Aut(X, B), p). Let Ctbl(X) denote the group of Borel automorphisms with countable support. It is shown that the topological group Aut 0(X, B ) = Aut(X, B)/Ctbl(X) is path-connected with respect to the quotient topology τ 0. It is also proved that Aut 0(X, B) has the Rokhlin property in the quotient topology p 0, i.e., the action of Aut 0(X,B) on itself by conjugation is topologically transitive.

Some applications of the Adams-Kechris technique

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every Σ_1^1 equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence relations and derive analogous results for the classification problem of Borel automorphisms.

Borel automorphisms with no finite invariant measure

An uncountable family of non-isomorphic Borel automorphisms which do not preserve any finite measure is presented.

The classification of hypersmooth Borel equivalence relations

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel auto-morphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism.

CAT(-1)-spaces, divergence groups and their commensurators

A CAT(−1)-space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane ([B], [BriHa], [Gr]). The CAT(−1)-property is one among many possible generalizations to singular spaces of the notion of negative curvature. Important examples of CAT(−1)-spaces include Riemannian manifolds of sectional curvature k ≤ −1 and their convex subsets ([B-G-S]), metric trees and piecewise hyperbolic cell complexes ([Mou],[Da],[Hag],[Be 1],[Be 2],[B-Br]). In this paper we establish certain superrigidity results for isometric actions of a group Λ on a CAT(−1)-space in the following two settings: A. The group Λ is a subgroup of a locally compact group G with Γ < Λ < ComGΓ, where Γ < G is a sufficiently large discrete subgroup and ComGΓ = {g ∈ G : g−1Γg and Γ share a subgroup of finite index} is the commensurator of Γ in G. B. The group Λ is an irreducible lattice in G := ∏n α=1Gα(kα), where each Gα is a semisimple algebraic group defined over a local field kα. The issues addressed in this paper are motivated on one hand by earlier work of G.A. Margulis ([Ma]) dealing with the linear representation theory of Λ, where in case A, G is a semisimple group and Γ < G a lattice, and on the other hand by the results of Lubotzky, Mozes and Zimmer ([L-M-Z]) concerning isometric actions of Λ on trees, where Γ < Λ < ComGΓ, G is the group of automorphisms of a regular tree and Γ < G is a lattice. Our approach to establishing superrigidity results is based on ergodic theoretic methods developed by Margulis ([Ma],[Zi 3],[A’C-B]). In this context, the following notion of boundary of a locally compact group Γ will be useful: let B be a standard Borel space on which Γ acts by Borel automorphisms preserving a σ-finite measure class μ.

The structure of hyperfinite Borel equivalence relations

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.

The Structure of Hyperfinite Borel Equivalence Relations

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.

Amenable versus hyperfinite Borel equivalence relations

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

The Full Group of a Countable Measurable Equivalence Relation

We study the group of all of a countable equivalence relation R on a standard Borel space, special Borel automorphisms whose graphs lie in R. We show that such a group always contains periodic maps of each order sufficient to generate R . A construction based on these periodic maps leads to totally nonperiodic R-automorphisms all of whose powers have disjoint graphs. The presence of a large number of periodic maps allows us to present a version of the Rohlin Lemma for R-automorphisms. Finally we show that this group always contains copies of free groups on any countable number of generators.

The full group of a countable measurable equivalence relation

We study the group of all " R R -automorphisms" of a countable equivalence relation R R on a standard Borel space, special Borel automorphisms whose graphs lie in R R . We show that such a group always contains periodic maps of each order sufficient to generate R R . A construction based on these periodic maps leads to totally nonperiodic R R -automorphisms all of whose powers have disjoint graphs. The presence of a large number of periodic maps allows us to present a version of the Rohlin Lemma for R R -automorphisms. Finally we show that this group always contains copies of free groups on any countable number of generators.

On the existence of a finite invariant measure

A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.