arXiv subjects

Xiangdong Ye

Publications and source records attributed to Xiangdong Ye.

58 records · Page 4Linked to original sources

arXiv2011

Local entropy theory for a countable discrete amenable group action

In the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action. For such an action, we introduce entropy tuples in both topological and measure-theoretic settings and build the variational relation between these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover. Moreover, based the idea of topological entropy pairs, we introduce and study two special classes of such an action: uniformly positive entropy and completely positive entropy. Note that in the building of the local variational principle, following Romagnoli's ideas two kinds of measure-theoretic entropy are introduced for finite Borel covers. These two kinds of entropy turn out to be the same, where Danilenko's orbital approach becomes an inevitable tool.

math.DS↗

arXiv2010

Family-independence for topological and measurable dynamics

For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.

math.DS↗

arXiv2010

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.

math.DS↗

arXiv2010

Product recurrent properties, disjointness and weak disjointness

Let $\F$ be a collection of subsets of $\Z_+$ and $(X,T)$ be a dynamical system. $x\in X$ is $\F$-recurrent if for each neighborhood $U$ of $x$, $\{n\in\Z_+:T^n x\in U\}\in \F$. $x$ is $\F$-product recurrent if $(x,y)$ is recurrent for any $\F$-recurrent point $y$ in any dynamical system $(Y,S)$. It is well known that $x$ is $\{infinite\}$-product recurrent if and only if it is minimal and distal. In this paper it is proved that the closure of a $\{syndetic\}$-product recurrent point (i.e. weakly product recurrent point) has a dense minimal points; and a $\{piecewise syndetic\}$-product recurrent point is minimal. Results on product recurrence when the closure of an $\F$-recurrent point has zero entropy are obtained. It is shown that if a transitive system is disjoint from all minimal systems, then each transitive point is weakly product recurrent. Moreover, it proved that each weakly mixing system with dense minimal points is disjoint from all minimal PI systems; and each weakly mixing system with a dense set of distal points or an $\F_s$-independent system is disjoint from all minimal systems. Results on weak disjointness are described when considering disjointness.

math.DS↗