Entropy theory without a past

Abstract

This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts the relative disjointness in the sense of Furstenberg, of 0-entropy extensions from completely positive entropy (c.p.e.) extensions. This theorem is used to prove several classical results in the general setup. E.g. we show that the Pinsker factor of a product system is equal to the product of the Pinsker factors of the component systems. Another application is to obtain a generalization (as well as a simpler proof) of the quasifactor theorem for 0-entropy systems of [GW].

Multidimensional sofic shifts without separation and their factors

“…We have factor maps of Z d measure-preserving systems, κ : (X, S, μ) → (X , S , μ ) and φ : (X, S, μ) → (Y, T, μ). Because one factor is zero entropy and the other is c.p.e., the disjointness theorem of [10] tells us that the map κ × φ sends μ onto the product measure μ × μ. Now consider the decomposition of μ relative to μ , which for μ -almost all x in X assigns a measure μ x to the fiber κ −1 (x ).…”

“…In some cases, our examples can be arranged to have topologically completely positive entropy, but at the same time to have no factor admitting an invariant measure such that as a measurable system it is of completely positive entropy. The proof (6.1) of the latter property appeals to a Z d disjointness theorem of Glasner, Thouvenot and Weiss [10].…”

The Spectrum of Completely Positive Entropy Actions of Countable Amenable Groups

“…Let us recall some results and definitions concerning the entropy of an action T of a countable discrete amenable group G on a Lebesgue space ðX ; BðX Þ; mÞ (see [6,13]). If P is a finite partition of X and F a subset of G;…”

“…Recall the definition of the Pinsker algebra PðTÞ of the action T of G on ðX ; BðX Þ; m X Þ: PðTÞ is the maximal (T-invariant) s-subalgebra of BðX Þ such that for any finite partition P & PðTÞ one has hðP; TÞ ¼ 0: The relative Pinsker algebra PðTjAÞ with respect to a T-invariant s-subalgebra A & BðX Þ can be introduced in a similar way [2,6,9,13,15].…”

Isomorphism rigidity of commuting automorphisms

“…As the Pinsker algebra of the product of two measure preserving Z d -actions is the product of their Pinsker algebras (cf. [11,Theorem 4]), and since X 2 has completely positive entropy, there exists a measurable map g : X → Y such that q(x, x ) = g(x) almost everywhere with respect to λ H . An application of Fubini's theorem shows that for λ H -a.e.…”