Dynamics of self-maps in their primal topologies
Jose C. Martin
TL;DR
The paper investigates the dynamics of a self-map $f$ in the primal topology $\tau_f$ induced by $f$, situating the analysis within Alexandroff spaces and their quasi-uniformities. It proves that any continuous map on an Alexandroff space is Lyapunov stable under the induced quasi-uniformity, ruling out Auslander-Yorke and Devaney chaos in this setting. On the primal space, the non-wandering set is $\Omega(f)=\bigcap_{k} f^k(X)$, the periodic, limit, and recurrent points coincide, minimal sets are precisely periodic orbits, and transitivity occurs only when $X$ is a single periodic orbit. The paper also establishes that topological ergodicity, weak and strong mixing are equivalent, and it characterizes proximality in the finest quasi-uniformity, including a decomposition of $X$ into connected components governed by orbit structure, with $X$ expressible as $\mathcal{O}(x)\cup\hat{x}(\mathbb{N})$ for some infinite preorbit. These results illuminate how primal Alexandroff topology imposes a clean, orbit-centric picture of dynamics and yields precise classifications for non-$T_1$ systems.
Abstract
We study a series of dynamical concepts for self-maps in the primal topology induced by them. Among the concepts studied are non-wandering points, limit points, recurrent points, minimal sets, transitive points and self-maps, topologically ergodic self-maps, weakly mixing self-maps, strongly mixing self-maps, Lyapunov stable self-maps, chaotic self-maps in the sense of Auslander-Yorke, chaotic self-maps in the sense of Devaney, asymptotic pairs, proximal pairs, and syndetically proximal pairs. Some results are given in the more general context of continuous self-maps in an Alexandroff topological space. We prove that a continuous self-map of an Alexandroff space is always Lyapunov stable.