Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMean equicontinuity, almost automorphy and regularity
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 a minimal topological dynamical system has trivial (one point) fibres. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibres have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.
rate research
Read More
In this paper, it is shown that for a minimal system $(X,T)$ and $d,kin mathbb{N}$, if $(x,x_i)$ is regionally proximal of order $d$ for $1leq ileq k$, then $(x,x_1,ldots,x_k)$ is $(k+1)$-regionally proximal of order $d$. Meanwhile, we introduce the notion of $mathrm{IN}^{[d]}$-pair: for a dynamical system $(X,T)$ and $din mathbb{N}$, a pair $(x_0,x_1)in Xtimes X$ is called an $mathrm{IN}^{[d]}$-pair if for any $kin mathbb{N}$ and any neighborhoods $U_0 ,U_1 $ of $x_0$ and $x_1$ respectively, there exist integers $p_j^{(i)},1leq ileq k,$ $1leq jleq d$ such that $$ bigcup_{i=1}^k{ p_1^{(i)}epsilon(1)+ldots+p_d^{(i)} epsilon(d):epsilon(j)in {0,1},1leq jleq d}backslash {0}subset mathrm{Ind}(U_0,U_1), $$ where $mathrm{Ind}(U_0,U_1)$ denotes the collection of all independence sets for $(U_0,U_1)$. It turns out that for a minimal system, if it dose not contain any nontrivial $mathrm{IN}^{[d]}$-pair, then it is an almost one-to-one extension of its maximal factor of order $d$.
Let $(X,T)$ be a topological dynamical system, and $mathcal{F}$ be a family of subsets of $mathbb{Z}_+$. $(X,T)$ is strongly $mathcal{F}$-sensitive, if there is $delta>0$ such that for each non-empty open subset $U$, there are $x,yin U$ with ${ninmathbb{Z}_+: d(T^nx,T^ny)>delta}inmathcal{F}$. Let $mathcal{F}_t$ (resp. $mathcal{F}_{ip}$, $mathcal{F}_{fip}$) be consisting of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets). The following Auslander-Yorkes type dichotomy theorems are obtained: (1) a minimal system is either strongly $mathcal{F}_{fip}$-sensitive or an almost one-to-one extension of its $infty$-step nilfactor. (2) a minimal system is either strongly $mathcal{F}_{ip}$-sensitive or an almost one-to-one extension of its maximal distal factor. (3) a minimal system is either strongly $mathcal{F}_{t}$-sensitive or a proximal extension of its maximal distal factor.
In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.
The weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x to mu_x^G$ is continuous, where $mu_x^G$ is the unique ergodic measure on ${ol{Orb(x)}, G}$.
We study dynamical systems that have bounded complexity with respect to three kinds of directional metrics: the directional Bowen metric $d_k^{vec{v},b}$, the directional max-mean metric $hat{d}_k^{vec{v},b}$ and the directional mean metric $overline{d}_k^{vec{v},b}$. It is shown that a $mathbb{Z}^q$-topological system $(X,T)$ has bounded topological complexity with respect to ${d_k^{vec{v},b}}$ (respectively ${hat{d}_k^{vec{v},b}}$) if and only if $T$ is $(vec{v},b)$-equicontinuous (respectively $(vec{v},b)$-equicontinuous in the mean). It turns out that a measure $mu$ has bounded complexity with respect to ${d_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-equicontinuous. Whats more, it is shown that $mu$ has bounded complexity with respect to ${overline{d}_k^{vec{v},b}}$ if and only if $mu$ has bounded complexity with respect to ${hat{d}_k^{vec{v},b}}$ if and only if $T$ is $(mu,vec{v},b)$-mean equicontinuous if and only if $T$ is $(mu,vec{v},b)$-equicontinuous in the mean if and only if $mu$ has $vec{v}$-discrete spectrum.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
