Nowhere dense competing holes in open dynamical systems

TL;DR

This paper investigates the behavior of trajectories in open dynamical systems on compact metric spaces, showing that the distribution of certain points influences whether trajectories visit shrinking neighborhoods infinitely often.

Contribution

It establishes that in systems with a nowhere dense set of centers, typical trajectories visit each shrinking neighborhood infinitely often, contrasting with the dense case.

Findings

Trajectories visit each neighborhood infinitely often when centers are nowhere dense.

The behavior differs significantly if the centers are somewhere dense.

Results depend on the topological structure of the set of centers.

Abstract

Let $\mathcal{M}$ be a compact metric space with no isolated points, and $f:\mathcal{M}\longrightarrow\mathcal{M}$ a homeomorphism. Consider a sequence of shrinking open balls $\{B^i_n\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$ with centers $\{p_i\}_{i=1}^\infty\subseteq\mathcal{M}$ and radii $\{\rho^i_n\}_{n=1}^\infty$. For every point $x\in\mathcal{M}$ and $n\in\mathbb{N}$, consider which ball the trajectory $\{x,f(x),f^2(x),\dots\}$ of the point first visits. We find that whenever the closure of $\{p_i\}_{i=1}^\infty$ is nowhere dense, and with very minor restrictions on $\{\rho_n^i\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$, the typical trajectory $\{f^k(x)\}_{k=0}^\infty$ will first visit, for each $i$, the ball $B^i_n$, for infinitely many $n$. This is never the case, should $\{p_i\}_{i=1}^\infty$ be somewhere dense. Keywords: Open Dynamical System, Topological Dynamics, Transitive Homeomorphism, Baire category. MSC2020: 37B05, 37B20, 18F60, 54E52.