Dimension theory for multimodal maps

TL;DR

This paper investigates the dimension theory and multifractal analysis of multimodal maps, extending existing results by describing the Lyapunov spectrum and exploring phase transitions in the spectra.

Contribution

It generalizes previous work by Todd on the Lyapunov spectrum and analyzes the multifractal spectrum of pointwise dimension for multimodal maps.

Findings

Lyapunov spectrum described for multimodal maps

Multifractal spectrum of pointwise dimension analyzed

Phase transitions in spectra due to irregular thermodynamic formalism

Abstract

This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum, generalising previous results by Todd. We also study the multifractal spectrum of pointwise dimension. The lack of regularity of the thermodynamic formalism for this class of maps is reflected in the phase transitions of the spectra.