Soficity, amenability, and dynamical entropy

There is no compact metrizable space containing all continua as unique components

We define a subgroup of the universal sofic group, obtained as the normalizer of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.

Let (X, Ɓ, m) be a standard probability space and let T be an ergodic non-singular automorphism on (X, Ɓ, m). A complex number λ is said to an L ∞ eigenvalue of T if there is a non-zero function f λ ∈ L ∞(X, Ɓ, m) such that f λ (Tx) = λf(x) a.e. m. We call any such f λ an L ∞ eigenfunction of T corresponding to the eigenvalue λ. Since ||f λ ∘ T||∞ = ||f λ ||∞ we have |λ| = 1. The collection e(T) of L ∞ eigenvalues of T forms a subgroup of the circle group. Further $$\left| {{f_\lambda }\left( {Tx} \right)} \right| = \left| \lambda \right|\left| {{f_\lambda }\left( x \right)} \right| = \left| {{f_\lambda }\left( x \right)} \right|a.e.m.$$ Since T is ergodic |f λ | is constant a.e. m. The function \(\frac{{{f_\lambda }}}{{\left| f \right|}}\) is an eigenfunction of absolute value one, with eigenvalue λ

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $L^1$ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor. The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon--Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

Consider a free ergodic measure-preserving profinite action Γ↷X (i.e., an inverse limit of actions Γ↷Xn, with Xn finite) of a countable property (T) group Γ (more generally, of a group Γ which admits an infinite normal subgroup Γ0 such that the inclusion Γ0⊂Γ has relative property (T) and Γ/Γ0 is finitely generated) on a standard probability space X. We prove that if w:Γ×X→Λ is a measurable cocycle with values in a countable group Λ, then w is cohomologous to a cocycle w′ which factors through the map Γ×X→Γ×Xn, for some n. As a corollary, we show that any orbit equivalence of Γ↷X with any free ergodic measure-preserving action Λ↷Y comes from a (virtual) conjugacy of actions.

We find conditions which guarantee that a given flow Φ on a compact metric space X admits a Lyapunov 1-form ω lying in a prescribed Cech cohomology class ξ∈H ^ 1 (X;ℝ). These conditions are formulated in terms of the restriction of ξ to the chain recurrent set of Φ. The result of the paper may be viewed as a generalization of a well-known theorem by Conley about the existence of Lyapunov functions.

For a compact space K we denote by $$C_w(K)$$ ( $$C_p(K)$$ ) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can $$C_w(K)$$ and $$C_p(K)$$ be homeomorphic? We show that the answer is “no”, provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K.

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\mu) is generically turbulent.

For a given (generally, assumed atomic) nontrivial Boolean algebra B and any a, b in B with b ≠ ∅, a conditional event (a|b) is said to exist if for all probability measures P over B, there exists some mapping P^(not necessarily a probability measure) and some algebra B^ (not necessarily Boolean) containing (a|b) such that if P(b) > 0, then P^((a|b)) = P(a|b), ordinary conditional probability of a given b. But, in turn, Lewis’ famous triviality result shows equivalently that if his forcing hypothesis holds—i.e., B and B^ at least overlap at B*(a, b, (a|b)), the Boolean subalgebra generated by a, b, and (a|b) (and for consistency P = P^)—then P must be trivial in the sense that a, b must be P-independent. The apparent informal implication of this, as gleaned from Lewis’ own comments and that of the numerous papers that are based on his result, is that this spells the “death knell” for Stalnaker’s thesis or even for any attempt at constructing a consistent nontrivial algebra of conditional events deriving from a Boolean algebra B. In this paper, we show two basic results contradicting this “conclusion”. First, fully consistent and nontrivial Boolean (in fact, sigma-) algebras of conditional events can always be constructed relative to any given Boolean algebra B, based upon a standard product probability space construction. This avoids Lewis’ forcing hypothesis—and thus Lewis triviality—yet satisfies natural isomorphic and homomorphic relations with B. Second, a key step in Lewis’ own proof, involving conditioning, when generalized to a setting where his forcing hypothesis no longer is imposed, leads directly to a simple criterion determining the structure of consistent nontrivial Boolean conditional event algebras. This criterion is also directly related to the well-known property of import/export in AI (Artificial Intelligence) systems and conditional logic.

Nous établissons que, pour tout groupe dénombrable abélien Γ et tout sous-groupe finiment engendré Δ de Γ, l’ensemble des actions de Δ sur un espace de probabilités standard (X,μ) qui peuvent être étendues en une action libre de Γ sur (X,μ) est générique (au sens de Baire). Ce résultat étend un théorème d’Ageev, qui correspond au cas où Δ est un groupe cyclique infini.

Supercompact minus compact is super

We prove that a compact space embeds into a ‐product of compact metrizable spaces (‐product of intervals) if and only if is (strongly countable‐dimensional) hereditarily metalindelöf and every subspace of has a nonempty relative open second countable subset. This provides novel characterizations of ‐Corson and compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the ‐product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space , the property of being compact is determined by the topological structure of the space of continuous real‐valued functions of equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.

Continua in the Gromov–Hausdorff space

Cell-like resolutions in the strongly countable [formula omitted]-dimensional case

Let π : Y → X \pi : Y\rightarrow X be a continuous surjection between compact Hausdorff spaces Y Y and X X which is irreducible in the sense that if F ⊊ Y F\subsetneq Y is closed, then π ( F ) ≠ X \pi (F)\neq X . We exhibit isomorphisms between various Boolean algebras associated to this data: the regular open sets of X X , the regular open sets of Y Y , the regular ideals of C ( X ) C(X) and the regular ideals of C ( Y ) C(Y) . We call X X and Y Y Boolean equivalent if the regular open sets of X X and the regular open sets of Y Y are isomorphic Boolean algebras. We give a characterization of when two compact metrizable spaces are Boolean equivalent; this characterization may be viewed as a topological version of the characterization of standard Borel spaces.