First hyperbolic times for intermittent maps with unbounded derivative

TL;DR

This paper investigates the statistical properties of hyperbolic times in a class of nonuniformly expanding maps with unbounded derivatives, linking hyperbolic time distributions to correlation decay and central limit theorems.

Contribution

It provides sharp estimates on hyperbolic time distributions for maps with unbounded derivatives, connecting these to decay of correlations and CLT, using a geometric approach and large deviations.

Findings

Sharp estimates on the distribution of first hyperbolic times.

Comparison of hyperbolic times approach with Young tower estimates.

Maps exhibit intermittent dynamics with unbounded derivatives.

Abstract

We establish some statistical properties of the hyperbolic times for a class of nonuniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric Baker's map type construction, exhibit intermittent dynamics, and have unbounded derivatives. The geometric approach captures various examples from the literature over the last thirty years. The statistics of these maps are controlled by the order of tangency that a certain "cut function" makes with the boundary of the square. Using a large deviations result of Melbourne and Nicol we obtain sharp estimates on the distribution of first hyperbolic times. As shown by Alves, Viana and others, knowledge of the tail of the distribution of first hyperbolic times leads to estimates on the rate of decay of correlations and derivation of a CLT. For our family of maps, we compare the estimates on correlation decay rate and CLT derived via hyperbolic times with those derived by a direct Young tower construction. The latter estimates are known to be sharp.