Related papers: A monotone Sinai theorem

A classical fact in ergodic theory is that ergodicity is equivalent to almost everywhere divergence of ergodic sums of all nonnegative integrable functions which are not identically zero. We show two methods, one in the measure preserving…

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show…

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the…

A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques…

We introduce a novel entropy-related function, \textit{non-repeatability}, designed to capture dynamical behaviors in complex systems. Its normalized form, \textit{mutability}, has been previously applied in statistical physics as a…

We prove a model theorem for factor maps between ergodic, infinite measure-preserving systems.

Extending to all probability measures the notion of m-equicontinuous cellular automata introduced for Bernoulli measures by Gilman, we show that the entropy is null if m is an invariant measure and that the sequence of image measures of a…

Stochastic monotonicity is a well known partial order relation between probability measures defined on the same partially ordered set. Strassen Theorem establishes equivalence between stochastic monotonicity and the existence of a coupling…

In this paper it is proved that if a minimal system has the property that its sequence entropy is uniformly bounded for all sequences, then it has only finitely many ergodic measures and is an almost finite to one extension of its maximal…

Let $p$ and $q$ be probability vectors with the same entropy $h$. Denote by $B(p)$ the Bernoulli shift indexed by $\Z$ with marginal distribution $p$. Suppose that $\phi$ is a measure preserving homomorphism from $B(p)$ to $B(q)$. We prove…

Let $\Gamma$ be a sofic group with a copy of $\mathbb{Z}$ in its center. We construct an uncountable family of pairwise nonisomorphic measure-preserving $\Gamma$ actions with completely positive entropy, none of which is a factor of a…

An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In…

Let $G$ be a countable discrete sofic group. We define a concept of uniform mixing for measure-preserving $G$-actions and show that it implies completely positive sofic entropy. When $G$ contains an element of infinite order, we use this to…

Dini's Theorem guarantees that a monotone sequence of continuous functions converges pointwise on a compact interval to a continuous limit that converges uniformly. In this paper, we establish new theorems generalizing Dini's result by…

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This…

This article presents in a self-contained way A. Uhlmann's celebrated Theorem of monotonicity of the relative entropy under completely positive and trace preserving maps. The Theorem is presented in its more general form and meaningful…

We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence of probability spaces and a sequence of measure-preserving maps between these spaces. This notion generalizes the classical concept of metric…

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive…