Standard deviation of recurrence times for piecewise linear transformations

Authors

Mimoon Ismael, University of AnbarFollow
Rodney Nillsen, University of WollongongFollow
Graham H. Williams, University of WollongongFollow

RIS ID

114840

Publication Details

Ismael, M., Nillsen, R. & Williams, G. H. (2017). Standard deviation of recurrence times for piecewise linear transformations. Asian-European Journal of Mathematics, 10 (1), 1750009-1-1750009-10.

Abstract

This paper is concerned with dynamical systems of the form (X,f) where X is a bounded interval and f comes from a class of measure-preserving, piecewise linear transformations on X. If A⊆X is a Borel set and x∈A, the Poincaré recurrence time of x relative to A is defined to be the minimum of {n:n∈Nandfn(x)∈A}, if the minimum exists, and ∞ 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.