From thermodynamic and spectral phase transitions to multifractal analysis

TL;DR

This paper explores the connection between thermodynamic and spectral phase transitions in certain dynamical systems and how these transitions influence the multifractal analysis of the Lyapunov spectrum, especially in partially hyperbolic endomorphisms.

Contribution

It demonstrates that thermodynamic and spectral phase transitions imply multifractal analysis of the Lyapunov spectrum in specific partially hyperbolic endomorphisms.

Findings

Existence of thermodynamic phase transitions in certain non-uniformly expanding maps.

Spectral phase transitions characterized by the presence or absence of spectral gaps.

Multifractal analysis of the Lyapunov spectrum influenced by these phase transitions.

Abstract

It is known that all uniformly expanding or hyperbolic dynamics have no phase transition with respect to H\"older continuous potentials. In \cite{BC21}, is proved that for all transitive $C^{1+\alpha}-$local diffeomorphism $f$ on the circle, that is neither a uniformly expanding map nor invertible, has a unique thermodynamic phase transition with respect to the geometric potential, in other words, the topological pressure function $\mathbb{R} \ni t \mapsto P_{top}(f,-t\log|Df|)$ is analytic except at a point $t_{0} \in (0 , 1]$. Also it is proved spectral phase transitions, in other words, the transfer operator $\mathcal{L}_{f,-t\log|Df|}$ acting on the space of H\"older continuous functions, has the spectral gap property for all $t<t_0$ and does not have the spectral gap property for all $t\geq t_0$. Our goal is to prove that the results of thermodynamical and spectral phase transitions imply a multifractal analysis for the Lyapunov spectrum. In particular, we exhibit a class of partially hyperbolic endomorphisms that admit thermodynamical and spectral phase transitions with respect to the geometric potential, and we describe the multifractal analysis of your central Lyapunov spectrum.