Multifractal analysis of weak Gibbs measures for non-uniformly expanding C^1 maps

TL;DR

This paper studies the local dimension spectrum of weak Gibbs measures in non-uniformly hyperbolic systems, presenting multiple characterizations and proving spectrum analyticity under certain conditions.

Contribution

It extends the understanding of local dimension spectra to non-uniform hyperbolic systems, providing new representations and analyticity results.

Findings

Spectrum described via invariant measures

Spectrum characterized by ergodic measures and equilibrium states

Analyticity of the spectrum established under additional assumptions

Abstract

We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.