Cover times in dynamical systems

TL;DR

This paper investigates the cover time in one-dimensional dynamical systems, providing asymptotic formulas for the expected time for orbits to become dense, with applications to specific maps like the Gauss and Manneville-Pomeau maps.

Contribution

It introduces transfer operator and inducing techniques to derive asymptotic formulas for cover times in hyperbolic and non-hyperbolic systems, extending understanding of orbit density rates.

Findings

Asymptotic formula for expected cover time derived

Results apply to Gauss map and Manneville-Pomeau maps

Link between measure decay rate and cover time established

Abstract

Given a one-dimensional dynamical system we study its cover time, which quantifies the rate at which orbits become dense in the state space. Using transfer operator tools for dynamical systems with holes and inducing techniques, for a wide class of uniformly hyperbolic and non-uniformly hyperbolic systems we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. Applications include the Gauss map and Manneville-Pomeau maps.