Standard deviation of recurrence times for piecewise linear transformations

Abstract

This paper is concerned with dynamical systems of the form [Formula: see text], where [Formula: see text] is a bounded interval and [Formula: see text] comes from a class of measure-preserving, piecewise linear transformations on [Formula: see text]. If [Formula: see text] is a Borel set and [Formula: see text], the Poincaré recurrence time of [Formula: see text] relative to [Formula: see text] is defined to be the minimum of [Formula: see text], if the minimum exists, and [Formula: see text] otherwise. The mean of the recurrence time is finite and is given by Kac’s recurrence formula. In general, the standard deviation of the recurrence times need not be finite but, for the systems considered here, a bound for the standard deviation is derived.