Almost sure convergence of cover times for $ψ$-mixing systems
Boyuan Zhao
TL;DR
This work analyzes almost-sure cover times for repellers of $\\psi$-mixing dynamical systems, introducing the stretched Minkowski dimension to handle non-doubling measures. It shows that typical cover-time growth is governed by the (stretched) Minkowski dimensions: finite upper Minkowski dimension yields exact limsup/liminf scaling with classical dimensions under exponential $\\psi$-mixing, while non-mixing requires weaker lower bounds and can fail to meet those limits. The authors provide concrete examples, including finitely and infinitely full-branched affine maps, to illustrate when stretched dimensions are essential, and they compute explicit behavior for irrational rotations as counterexamples. They also extend the framework to flows via suspension constructions, establishing analogous dimension-based criteria (with a unit-dimension shift) and confirming that the flow dimension satisfies $\\dim_M(\\nu)=\\dim_M(\\mu)+1$ in suspension examples. Overall, the paper clarifies how geometric-dimension concepts govern rare-event cover times in chaotic, mixing systems and broadens the applicability to flows and non-doubling measures.
Abstract
Given a topologically transitive system on the unit interval, one can investigate the cover time, i.e. time for an orbit to reach certain level of resolution in the repeller. We introduce a new notion of dimension, namely the stretched Minkowski dimension, and show that under mixing conditions, the asymptotics of typical cover times are determined by Minkowski dimensions when they are finite, or by stretched Minkowski dimensions otherwise. For application, we show that for countably full-branched affine maps, results using the usual Minkowski dimensions fail to produce a finite log limit of cover times whilst the stretched version gives an finite limit. In addition, cover times of irrational rotations are explicitly calculated as counterexamples, due to the absence of mixing.