In this chapter we analyze the structure of the quotient of the action space by the relation of weak equivalence. In this chapter, we introduce a topology on the action space modulo weak equivalence that is finer than that previously studied in the literature.
Uniform mixing and completely positive sofic entropy
This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor group has sophic entropy at most zero. We note that a distal systemΓ has zero naive entropy in both senses if it has an element of infinite order.
Weak equivalence of measurable group actions
INVARIANT RANDOM SUBGROUPS AND ACTION VERSUS REPRESENTATION MAXIMALITY
Introduction
We recall that for a ∈ A(G,X, µ), we have a b if and only if a lies in the closure of isomorphic copies of b. Since the isomorphic copies of sα are dense in A(G, X, µ), this implies that the isomorphic copies of τσβ×ι are dense in A(G,X, µ).
Proof of Theorem 1.1.2
By [74, Theorem 3.11] this implies that any ergodic action d of N is weakly contained in almost every ergodic component of c. It is possible that one could use the techniques developed in this paper to show that Theorem 1.1.2 also holds for the free groups with finitely many generatorsn> 1 but we have not verified this.
TOPOLOGY AND CONVEXITY IN THE SPACE OF ACTIONS MODULO WEAK EQUIVALENCE
Introduction
Fritz's objects are referred to as 'convex spaces'; we weaken the definition to include the convex structure of A∼(Γ,X, µ), obtaining the notion of 'weakly convex space'. Tucker-Drob and Bowen have also shown that A∼s(Γ,X, µ) is a simplex and the set FR∼s(Γ,X, µ) of stable weak equivalence classes of free actions is a subsimplex.
Weak convex spaces
We also thank Robin Tucker-Drob for informing us of his result with Bowen that the space of stable weak equivalence classes forms a simplex, and for raising the question of when it forms a Poulsen simplex. We can define extreme points in a weakly convex space in exactly the same way as in a vector space.
Topology on the space of weak equivalence classes
1be the finite partition of K given by the atoms of the Boolean algebra created by (Aiγ)i≤N,γ∈Fi. Since the construction is independent of the set chosen to realize L, we actually have dKL.
The space of weak equivalence classes as a weak convex space
Let A∼s(Γ,X, µ) be the space of stable weak equivalence classes and let ι be a trivial action of Γ on a standard probability space. So we can find the partition (Ds,i)ls=1ofY such that for allp ≤ mand alls,t ≤ l we have We now write down a lemma that will be useful later and guarantees that the metric on A∼(Γ,X, µ) behaves nicely with respect to the convex structure.
It is in fact possible to define integrals of weak equivalence classes of actions over a probability measure. Suppose(Z, η) is a probability space and suppose that for each zw there is a probability space(Yz, νz) and a measure-preserving actionΓ yaz (Yz, νz) such that the map z 7→ [az]from(Z, η) to A∗∼(Γ) is measurable, where[az] is the weak equivalence class ofaz. Y will be a standard probability space isomorphic to (X, µ) if (Z, η) is standard or η - almost all (Yz, νz) are standard.
The structure of the space of weak equivalence classes for amenable acting groups
Suppose that a,b ∈ A(Γ,X, µ) are actions such that type(a) = type(b) and are concentrated on subgroups with finite index in Γ. We can assume that θ = type(a)=type(b) is concentrated on subgroups of index n for some fixed n. Every transitive homomorphism φ : Γ → Sym(n) defines the conjugacy class Hφ of subgroups of index n of Γ as stabilizers of jφ.
For each a-orbit [x]Ea, the stabilizers of the action of Γ on [x]Ea also determine a conjugation class Hxa of index n subgroups of Γ. Let L be the set of all transitive homomorphisms φ: Γ → Sym(n) such that φ is @-least in[φ]Ec. In [74] Tucker-Drob shows that for acceptable Γ the space A∼s(Γ,X, µ) of stable weak equivalence classes is homeomorphic to the space IRS(Γ) of invariant random subgroups of Γ.
The structure of the space of weak equivalence classes for general acting groups
1 of partitions of Y such that for any Borel probability measure ρ onY, (Am)∞m=1 is dense in the set of k-partitions of X with topology inherited from MALG(Y, ρ). We can then find an index and a set A6 ⊆ A5 of positive measure such that for all z ∈ K, Hi = H separates xz from Cd(u) for allu ∈ G. Choose a finite collection Lof measurable subsets of X×2FN , so that for every measurable partition A of X×2FN there is a partitionB ⊆ L such that ρ.
Since c ≺ b, it is sufficient to show that for every partition A of X×2FN there is a partition C of X such that ρ. Ak) of X × 2FN there is a partition B whose pieces are disjoint unions of sets of the form Pθbforθ : F →r ×2 such that ρ. We note that the proof of Theorem 2.1.4 holds for any groupΓ such that an arbitrary free action in the uniform topology can be approximated by ergodic operations - for example any group of the form Z∗H.
The space of stable weak equivalence classes
Can every free action of Γ be approximated in the uniform topology of A(Γ,X, µ) by ergodic actions. The extreme points of A∼s(Γ,X, µ) are precisely those stable weak equivalence classes that contain an ergodic action. Thus, if a stable weak equivalence class contains an ergodic action, it is an extreme point of A∼s(Γ,X, µ).
On the other hand, an argument identical to the proof of Theorem 2.1.3 shows that if the stable weak equivalence class of an action a is an extreme point of A∼s(Γ,X, µ), then if we write a= ∫.
A TOPOLOGICAL SEMIGROUP STRUCTURE ON THE SPACE OF ACTIONS MODULO WEAK EQUIVALENCE
Introduction
We would like to thank Alexander Kechris for introducing us to this topic and asking if the product is continuous.
Definition of the fine topology
The symmetric argument shows that if n≥ N, then for allk,Ct,k(an×bn) is contained in a sphere with radius around Ct,k(a×b) and thus the theorem is proved.
WEAK EQUIVALENCE OF STATIONARY ACTIONS AND THE ENTROPY REALIZATION PROBLEM
- Introduction
- A characterization of weak containment
- The space of weak equivalence classes
- Proof of Theorem 4.1.1
It is also better behaved from the point of view of descriptive group theory: in general there is no standard Borel structure on the set of isomorphism classes of stationary actions, while in Section 4.3 we will define a natural Polish topology on the set of weak. Equivalence classes of stationary sim-actions for each pair (G,m). The purpose of this paper is to establish the following theorem, which shows that the above problem can be considered as a problem for the structure of the space of weak equivalence classes. The Furstenberg entropy is an invariant of weak equivalence and reduces to a continuous function in the space of weak equivalence classes.
In this section we verify that one obtains an equivalent notion if one changes the definition of the weak constraint to allow shifts on both sides of the intersections. As in the case of mass conservation discussed in [23], the convex structure is better behaved if weakly consistent conservation relations are considered. Arguments from [23] go on to show that Statgs(G,m,X, μ) is isomorphic to a compact convex subset of a Banach space, and that its extreme points are precisely those stable weak equivalence classes containing an ergodic action. .
Sofic entropy
NAIVE ENTROPY OF DYNAMICAL SYSTEMS
Introduction
Here, htpnv is the naive topological entropy and htpΣ is the sophic entropy with respect to a sophic approximationΣ. An advantage of naive entropy is that in many cases it is easy to see that a system has zero naive entropy. For example, in Section 5.2 we observe that if Γ has an element of infinite order, then any distal Γ system has zero naive entropy in both senses.
Moreover, in Section 5.2 we can show that if Γ is a free group, the generic Γ system with phase space of the Cantor set has zero naive topological entropy. If this is a countable free group, then the set of F-systems with at most sophic entropy comes on top of F,2N. After reporting our results to Brandon Seward, he informed us that the measurable case of Bowen's naive entropy conjecture has been independently proven by a number of researchers, including Miklos Abert, Tim Austin, Seward himself, and Benjamin Weiss.
Naive entropy
Our work was performed independently of the (yet unpublished) work of these authors on the measurable case. If αand β are partitions of (X, µ), the connection α∨ β is the partition consisting of all intersections A∩Bwhere A ∈ αand B ∈ β. We now introduce two standard reformulations of the definition of naive topological entropy, originally due in the case of Zto R.
The proof of the following is an immediate generalization of the corresponding statement for Z-systems, which can be found as Proposition 14.11 in [42]. B(γas) is contained in the element UF. 5.2) If V is an open cover of X, let diam(V) denote the supremum of the diameters of the elements of V. The following argument is a direct generalization of the corresponding proof for Z-systems given as Part I of Theorem 17.1 in [ 42].
Sofic groups and sofic entropy
By Corollary 2.5 in [41], the set of homeomorphisms with zero entropy is uniformly dense in Homeo 2N. In this subsection, we find a large-separated subset V of Map(σ,F, δ) such that each element of V is approximately equivariant on a fixed large subset of[n]. In this subsection, we use the data constructed earlier to bound the size of a suitably separated subset of Map(σ,F, δ) in terms of the separation numbers used to calculate naive entropy.
Let i ≤ j and take a 2-voltage set Vi orV of minimal cardinality with respect to the pseudometric dB∞. Note that ifφ ∈V, then by the hypothesis that Vi is 2-overvoltage forV with respect to the metric dB∞. Since V is a subset of D and we have assumed that D is separated with respect tod∞, it follows that f is injective.
UNIFORM MIXING AND COMPLETELY POSITIVE SOFIC ENTROPY
- Introduction
- Preliminaries Notation
- Metrics on sofic approximations and uniform model-mixing
- Proof of Theorem 6.1.2
- Proof of Theorem 6.1.3
- Proof of Theorem 6.1.1
This has been done to avoid confusion with an alternative use of the word 'extinguished' in the physics literature. 6.1) Now let µnandVn be as in the statement of the lemma. withF = AVn andE = Snwe have lim inf. If v and w are in the same connected component of Hσ, then let ρσ be the W-weighted graph distance between v and w.
If (AG, µ) is mixing uniformly, then this sequence of marginals is clearly pattern-mixing uniform in the sense of Definition 6.3.1. Since also pn = o(Vn), it follows that σng·v lies on the same path as σng0 ·v with high probability inv. Note that the restriction of the action G on H is a permuted power of the original process Z in the sense of Definition 6.5 from [32].
BIBLIOGRAPHY
