Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence

By proving the minimality of face transformations acting on the diagonal points and searching the points allowed in the minimal sets, it is shown that the regionally proximal relation of order $d$, \RP^{[d]}, is an equivalence relation for minimal systems. Moreover, the lifting of \RP^{[d]} between two minimal systems is obtained, which implies that the factor induced by \RP^{[d]} is the maximal $d$-step nilfactor. The above results extend the same conclusions proved by Host, Kra and Maass for minimal distal systems. A combinatorial consequence is that if $S$ is a dynamically syndetic subset of $\Z$, then for each $d\ge 1$, \{(n_1,\...,n_d)\in \Z^d: n_1\ep_1+... +n_d\ep_d\in S, \ep_i\in \{0,1\}, 1\le i\le d\} is syndetic. In some sense this is the topological correspondence of the result obtained by Host and Kra for positive upper Banach density subsets using ergodic methods.Comment: 34 pages, 2 figures, the final version for submissio

Periodic points for amenable group actions on uniquely arcwise connected continua

We show that if $G$ is a countable amenable group acting on a uniquely arcwise connected continuum $X$, then $G$ has either a fixed point or a 2-periodic point in $X$

Sensitivity, proximal extension and higher order almost automorphy

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 $x,y\in U$ with $\{n\in\mathbb{Z}_+: d(T^nx,T^ny)>\delta\}\in\mathcal{F}$. 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-Yorke's 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.Comment: 24 pages, revised version following referees' reports. To appear in Transactions of the AM

A cubic nonconventional ergodic average with M\"obius and Liouville weight

It is shown that the cubic nonconventional ergodic average of order 2 with M\"obius and Liouville weight converge almost surely to zero. As a consequence, we obtain that the Ces\`aro mean of the self-correlations and some moving average of the self-correlations of M\"obius and Liouville functions converge to zero.Comment: In this version, we put in the surface our main result on the Cesaro mean of the auto-correlation of M\"obius and Liouville which is related to the very recent results of K. Matom\"aki and M. Radziwi{\l}{\l}, and K. Matom\"aki, M. Radziwi{\l}{\l} and T. Ta

Nil Bohr-sets and almost automorphy of higher order

Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can be viewed as the higher order version of an old question concerning Bohr sets is studied: for any $d\in {\mathbb N}$ does the collection of $\{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn)\neq \emptyset\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets? It is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem is answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\supset \{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn)\neq \emptyset\}.$ Moreover, it is shown that the answer of the other side of the problem can be deduced from some result by Bergelson-Host-Kra if modulo a set with zero density. In the second part, the notion of $d$-step almost automorphic systems with $d\in{\mathbb N}\cup\{\infty\}$ is introduced and investigated, which is the generalization of the classical almost automorphic ones. It is worth to mention that some results concerning higher order Bohr sets will be applied to the investigation. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$ is $d$-step almost automorphic can be characterized via various subsets of ${\mathbb Z}$ including the dual sets of $d$-step Poincar\'e and Birkhoff recurrence sets, and Nil$_d$ Bohr$_0$-sets. Moreover, it turns out that the condition $(x,y)\in X\times X$ is regionally proximal of order $d$ can also be characterized via various subsets of ${\mathbb Z}$.Comment: This paper consists of the following two papers arXiv:1109.3636 and arXiv:1110.6599. It will be published in "Memoirs of the AMS

Weakly mixing, proximal topological models for ergodic systems and applications

In this paper it is shown that every non-periodic ergodic system has two topologically weakly mixing, fully supported models: one is non-minimal but has a dense set of minimal points; and the other one is proximal. Also for independent interests, for a given Kakutani-Rokhlin tower with relatively prime column heights, it is demonstrated how to get a new taller Kakutani-Rokhlin tower with same property, which can be used in Weiss's proof of the Jewett-Krieger's theorem and the proofs of our theorems. Applications of the results are given

An answer to Furstenberg's problem on topological disjointness

In this paper we give an answer to Furstenberg's problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any nonempty open subset $U\subset X$, there is $x\in D\cap U$ satisfying that $\{n\in{ \mathbb Z}_+: T^nx\in U, S^ny\in V\}$ is syndetic. Some characterization for the general case is also described. As applications we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^n,T^{(n)})$ and $(X, T^n)$ for any $n\in { \mathbb N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_K)$ is disjoint from all minimal systems.Comment: To appear in Ergodic Theory and Dynamical System

Recurrence properties and disjointness on the induced spaces

A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability space. The connection among some dynamical properties on the original space and on the induced spaces are investigated. Particularly, a minimal weakly mixing system which induces a $P$-system on the probability space is constructed and some disjointness result is obtained.Comment: 16 pages, 2 table

Mean equicontinuity and mean sensitivity

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the later property must have zero entropy.Comment: 25 pages, changes suggested by the referee incorporated, to appear in Ergodic Theory and Dynamical System

Topology and Topological Sequence Entropy

Let $X$ be a compact metric space and $T:X\longrightarrow X$ be continuous. Let $h^*(T)$ be the supremum of topological sequence entropies of $T$ over all subsequences of $\mathbb Z_+$ and $S(X)$ be the set of the values $h^*(T)$ for all continuous maps $T$ on $X$. It is known that $\{0\} \subseteq S(X)\subseteq \{0, \log 2, \log 3, \ldots\}\cup \{\infty\}$. Only three possibilities for $S(X)$ have been observed so far, namely $S(X)=\{0\}$, $S(X)=\{0,\log2, \infty\}$ and $S(X)=\{0, \log 2, \log 3, \ldots\}\cup \{\infty\}$. In this paper we completely solve the problem of finding all possibilities for $S(X)$ by showing that in fact for every set $\{0\} \subseteq A \subseteq \{0, \log 2, \log 3, \ldots\}\cup \{\infty\}$ there exists a one-dimensional continuum $X_A$ with $S(X_A) = A$. In the construction of $X_A$ we use Cook continua. This is apparently the first application of these very rigid continua in dynamics. We further show that the same result is true if one considers only homeomorphisms rather than con\-ti\-nuous maps. The problem for group actions is also addressed. For some class of group actions (by homeomorphisms) we provide an analogous result, but in full generality this problem remains open. The result works also for an analogous class of semigroup actions (by continuous maps).Comment: 90 pages, the paper has been accepted for publication in SCIENCE CHINA Mathematic