Almost automorphy of surjective semiflows on compact Hausdorff spaces

TL;DR

This paper investigates the almost automorphic behavior of surjective semiflows on compact Hausdorff spaces and provides a comprehensive proof of Veech's structure theorem for such flows.

Contribution

It introduces the concept of a.a. points in semiflows and offers a complete proof of Veech's structure theorem for a.a. flows.

Findings

Characterization of a.a. points in semiflows

Complete proof of Veech's structure theorem

Insights into the dynamics of a.a. semiflows

Abstract

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.