Nowhere dense competing holes in open dynamical systems

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.