Phase Transitions of the Multifractal Spectrum

TL;DR

This paper investigates the phase transitions in the multifractal spectrum of Gibbs measures on countable Markov shifts, using thermodynamic formalism and applying results to the Gauss map.

Contribution

It provides a detailed analysis of the non-analytic points in the multifractal spectrum for countable Markov shifts, extending the understanding of phase transitions in this context.

Findings

Identification of phase transition points in the multifractal spectrum.

Application of the theory to the Gauss map.

Use of pressure function analyticity to analyze spectrum regularity.

Abstract

We consider the multifractal analysis of the pointwise dimension for Gibbs measures on countable Markov shifts. Our paper analyses the set of non-analytic points or phase transitions of the multifractal spectrum. By Sarig's thermodynamic formalism for countable Markov shifts and Iommi's expression of the multifractal spectrum, we apply analyticity arguments on the pressure function for the countable shift. Finally, we apply our results about the phase transitions of the multifractal spectrum to the Gauss map.