Related papers: Topological Speedups

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of…

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show…

Let $\pi: (X,T)\rightarrow (Y,T)$ be a factor map of topological dynamics and $d\in {\mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological characteristic factor if there exists a dense $G_\delta$ set $X_0$ of $X$ such that for each…

A countable family of $*$-commuting surjective, non-injective local homeomorphisms of a compact Hausdorff space $X$ gives rise to an action $\theta$ of a countably generated, free abelian monoid $P$. For such a triple $(X,P,\theta)$, which…

We show that a topometric space $X$ is topometrically isomorphic to a type space of some continuous first-order theory if and only if $X$ is compact and has an open metric (i.e., satisfies that $\{p : d(p,U) < \varepsilon\}$ is open for…

This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological…

Let $(T,X)$ with phase mapping $(t,x)\mapsto tx$ be a semiflow on a compact $\textrm{T}_2$-space $X$ with phase semigroup $T$ such that $tX=X$ for each $t$ of $T$. An $x\in X$ is called an \textit{a.a. point} if $t_nx\to y, x_n^\prime\to…

It is widely known that when $X$ is compact Hausdorff, and when $T: X \to X$ and $f: X \to \mathbb{R}$ are continuous, \begin{equation*} P(T,f) = \sup_{\text{$\mu$: Radon probability}} \left( h_\mu(T) + \int f\, \mathrm{d}\mu \right),…

In the topological dynamical system $(X,T)$, a point $x$ simultaneously approximates a point $y$ if there exists a sequence $n_1$, $n_2$, ... of natural numbers for which $T^{n_i} x$, $T^{2n_i}x$, ..., $T^{k n_i} x$ all tend to $y$. In…

We prove that if $T: X \to X$ is a selfmap of a set $X$ such that $\bigcap \{T^{n}X: n\in N}\}$ is a one-point set, then the set $X$ can be endowed with a compact Hausdorff topology so that $T$ is continuous.

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their…

Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function.…

This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure…

By definition, a map quasiperiodic on a set $X$ if the map is conjugate to a pure rotation. Suppose we have a trajectory $(x_n)$ that we suspect is quasiperiodic. How do we determine if it is? In this paper we show how to compute the…

Birkhoff's theorem states that for an ergodic automorphism, the time averages converge to the space average. Given sequence $\psi(n)\to+0$, U. Krengel proved that for any ergodic automorphism there is an indicator such that the…

A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that $\eta_i:\X_i\to \X_i$ is a continuous proper map on a locally compact…

We describe the approximation of a continuous dynamical system on a p. l. manifold or Cantor set by a tractable system. A system is tractable when it has a finite number of chain components and, with respect to a given full background…

In this paper we obtain an almost sure invariance principle for convergent sequences of either Anosov diffeomorphisms or expanding maps on compact Riemannian manifolds and prove an ergodic stability result for such sequences. The sequences…

We introduce a framework for analyzing topological tipping in time-evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, homological, and higher-order…

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis…