Large deviation principle for Benedicks-Carleson quadratic maps

TL;DR

This paper establishes a level 2 large deviation principle for Benedicks-Carleson quadratic maps, providing a statistical framework for understanding fluctuations in dynamics far from equilibrium.

Contribution

It introduces a novel approach using induced Markov maps and towers to prove large deviation principles for quadratic maps with slow recurrence.

Findings

Proves a level 2 large deviation principle for Benedicks-Carleson quadratic maps.

Extends large deviations results to dynamics far from equilibrium.

Uses induced Markov maps to control recurrence and fluctuations.

Abstract

Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks & Carleson to build induced Markov maps of arbitrarily small scale and associated towers, to which the absolutely continuous measures can be lifted. These various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle, for the absolutely continuous measures. This result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.