WAITING TIME DISTRIBUTIONS FOR THE DOUBLING MAP

Abstract

The study of waiting times, also referred to as hitting times or return times, plays a central role in ergodic theory, probability theory, and the theory of dynamical systems. Given a measurable dynamical system and a target set of small measure, one is naturally led to ask how long a typical orbit needs to wait before entering this set for the first time. This problem has deep connections with recurrence theory, limit laws, symbolic dynamics, and extreme value theory.

A classical motivation originates from the Poincaré recurrence theorem, which guarantees that almost every point returns infinitely often to any set of positive measure. However, Poincaré recurrence is qualitative in nature and provides no information about the distribution or growth rate of return times. Quantitative refinements of recurrence phenomena were initiated by Kac, who established his celebrated formula relating the expected return time to the inverse of the measure of the set.