Related papers: The structure of tame minimal dynamical systems fo…

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \N$, or it is a "tame" topological space whose topology…

Let $(X,G)$ be a minimal equicontinuous dynamical system, where $X$ is a compact metric space and $G$ some topological group acting on $X$. Under very mild assumptions, we show that the class of regular almost automorphic extensions of…

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of the Stone-Cech compactification of the natural numbers, or it…

Given a dynamical system $(X, \Gamma)$, the corresponding crossed product $C^*$-algebra $C(X)\rtimes_{r}\Gamma$ is called reflecting, when every intermediate $C^*$-algebra $C^*_r(\Gamma)<\mathcal{A} < C(X)\rtimes_{r}\Gamma$ is of the form…

For a finite discrete topological space $X$ with at least two elements, a nonempty set $\Gamma$, and a map $\varphi:\Gamma\to\Gamma$, $\sigma_\varphi:X^\Gamma\to X^\Gamma$ with $\sigma_\varphi((x_\alpha)_{\alpha\in\Gamma})=…

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi$ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu$ on $X$ and $\nu$ on $Y$,…

In this paper we characterize tame dynamical systems and functions in terms of eventual non-sensitivity and eventual fragmentability. As a notable application we obtain a neat characterization of tame subshifts $X \subset \{0,1\}^{\mathbb…

We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three…

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…

We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable…

We consider an almost o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$ and its tame extension $\mathcal N=(N,<,+,0,\ldots)$. We demonstrate that the subset $\{x \in M^n\;|\; \mathcal N \models \Phi(x,a)\}$ of $M^n$…

We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $\Gamma$, we consider…

This paper presents reduction theorems for stability, attractivity, and asymptotic stability of compact subsets of the state space of a hybrid dynamical system. Given two closed sets $\Gamma_1 \subset \Gamma_2 \subset \Re^n$, with…

If $\pi:(X,T)\to(Z,S)$ is a topological factor map between uniquely ergodic topological dynamical systems, then $(X,T)$ is called an isomorphic extension of $(Z,S)$ if $\pi$ is also a measure-theoretic isomorphism. We consider the case when…

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid…

Let $X$ be a compact metrizable group and $\Gamma$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\Gamma)$ is surjunctive, i.e., every injective…

We introduce dynamical systems $(X,G)$ with closed relations $G$ on compact metric spaces $X$ and discuss different types of minimality of such dynamical systems, all of them generalizing minimal dynamical systems $(X,f)$ with continuous…

A big-isotropic structure $E$ is an isotropic subbundle of $TM\oplus T^*M$, endowed with the metric defined by pairing. The structure $E$ is said to be integrable if the Courant bracket $[\mathcal{X},\mathcal{Y}]\in\Gamma E$,…

We present a graph-theoretic model for dynamical systems $(X,\sigma)$ given by a surjective local homeomorphism $\sigma$ on a totally disconnected compact metrizable space $X$. In order to make the dynamics appear explicitly in the graph,…

Let $\bold G$ be a reductive algebraic group defined over $\Q$, and let $\Gamma$ be an arithmetic subgroup of $\bold G(\Q)$. Let $X$ be the symmetric space for $\bold G(\R)$, and assume $X$ is contractible. Then the cohomology (mod torsion)…