Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics

Let $\pi\colon T\times X\rightarrow X$ with phase map $(t,x)\mapsto tx$, denoted $(\pi,T,X)$, be a \textit{semiflow} on a compact Hausdorff space $X$ with phase semigroup $T$. If each $t\in T$ is onto, $(\pi,T,X)$ is called surjective; and if each $t\in T$ is 1-1 onto $(\pi,T,X)$ is called invertible and in latter case it induces $\pi^{-1}\colon X\times T\rightarrow X$ by $(x,t)\mapsto xt:=t^{-1}x$, denoted $(\pi^{-1},X,T)$. In this paper, we show that $(\pi,T,X)$ is equicontinuous surjective iff it is uniformly distal iff $(\pi^{-1},X,T)$ is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of $(\pi^{-1},X,T)$ if $(\pi,T,X)$ is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of $\mathbb{Z}$-flow with compact zero-dimensional phase space.

Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces

Almost automorphy and Veech's relations of dynamics on compact T2-spaces

Characterizations of relativized distal points of topological dynamical systems

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 $t_nx\to y, x_n^\prime\to x^\prime$ and $t_nx_n^\prime=y$ implies $x=x^\prime$ for every net $\{t_n\}$ in $T$. In this paper, we study the a.a. dynamics of $(T,X)$; and moreover, we present a complete proof of Veech's structure theorem for a.a. flows.

Almost automorphy of surjective semiflows on compact Hausdorff spaces

We generalize an important theorem of Fred Galvin from the Stone-Cech compactification βT of any discrete semigroup T to any compact Hausdorff right-topological semigroup with a dense topological center; and then apply it to Ellis’ semigroups to prove that a point is distal if and only if it is IP*-recurrent, for any semiflow (T;X) with arbitrary compact Hausdorff phase space X not necessarily metrizable and with arbitrary phase semigroup T not necessarily cancelable.

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HaiLan Liang

Papers

On Galvin’s theorem for compact Hausdorff right-topological semigroups with dense topological centers