Multifractal formalism for Benedicks-Carleson quadratic maps

TL;DR

This paper develops a multifractal formalism for a class of quadratic maps, relating the Hausdorff dimension of level sets to entropies and Lyapunov exponents, and proves the continuity of the spectrum.

Contribution

It introduces a formula linking Hausdorff dimension to dynamical invariants and constructs towers to estimate dimensions and establish large deviation principles.

Findings

The Birkhoff spectrum is continuous for the considered maps.

A formula relating Hausdorff dimension to entropy and Lyapunov exponents is derived.

A large deviation principle for empirical distributions is established.

Abstract

For a positive measure set of nonuniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given observable and consider the associated {\it Birkhoff spectrum} which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to "see" sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle for empirical distributions, with Lebesgue as a reference measure.