Birkhoff spectra for one-dimensional maps with some hyperbolicity

TL;DR

This paper analyzes the multifractal structure of one-dimensional smooth dynamical systems, linking Birkhoff averages, Hausdorff dimensions, and hyperbolic measures, revealing detailed geometric properties of invariant sets.

Contribution

It provides a new characterization of the Hausdorff dimension of level sets from Birkhoff averages using hyperbolic measures for topologically mixing $C^2$ maps.

Findings

Hausdorff dimension of level sets characterized by hyperbolic measures

Full Hausdorff dimension of the complement of quasi-regular points

Characterization of ergodic basins in terms of local dimensions

Abstract

We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.