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.