Upper large deviations bound for singular-hyperbolic attracting sets

TL;DR

This paper establishes exponential large deviation bounds for certain semiflows over non-uniformly expanding maps with singularities, with applications to singular-hyperbolic attracting sets and physical measures.

Contribution

It introduces a new exponential large deviation upper bound for suspension semiflows over maps with singularities, extending understanding of statistical properties in these systems.

Findings

Exponential large deviation upper bounds for observables on semiflows.

Existence of physical measures for piecewise expanding maps with singularities.

Exponential escape rates from measure-zero subsets.

Abstract

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities and/or discontinuities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/discontinuous set of the base map. To obtain this upper bound, we show that the base transformation exhibits exponential slow recurrence to the singular set. The results are applied to semiflows modeling singular-hyperbolic attracting sets of $C^2$ vector fields. As corollary of the methods we obtain result on the existence of physical measure for classes of piecewise $C^{1+}$ expanding maps of the interval with singularities and discontinuities. We are also able to obtain exponentially fast escape rates from subsets without full measure.