Quantitative covering and equidistribution

Abstract Covering (density of orbits) and equidistribution are fundamental properties of an ergodic dynamical system. In this expository note, we investigate the quantitative aspects of these properties for a simple model. In particular, we investigate how much of a single orbit we must observe before we can expect it to reach a certain density in the state space and for it to be almost equidistributed, in a certain sense. We explore more broadly how these questions regarding the statistics of individual dynamical orbits are closely related to the multifractal properties of the underlying ergodic measure.