Multifractal analysis for multimodal maps

Abstract

Given a multimodal interval map $f:I\to I$ and a Hölder potential $\varphi :I\to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $\varphi$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions.

Minor rewrites

References in corpus (3)

• Non-uniform hyperbolicity in complex dynamics
• Return time statistics for invariant measures for interval maps with positive Lyapunov exponent
• Multifractal analysis of Lyapunov exponent for the backward continued fraction map

Cited by in corpus (7)

• The thermodynamic approach to multifractal analysis
• Statistical stability of equilibrium states for interval maps
• Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets
• Topological Pressure for the Completely Irregular Set of Birkhoff Averages
• Multifractal analysis of the irregular set for almost-additive sequences via large deviations
• On the topological entropy of saturated sets for amenable group actions
• Multifractal formalism derived from thermodynamics