Lyapunov spectrum of asymptotically sub-additive potentials

TL;DR

This paper investigates the multifractal analysis of asymptotically sub-additive potentials in dynamical systems, establishing variational principles relating entropy and pressure without requiring unique equilibrium measures or pressure differentiability.

Contribution

It introduces new variational relations for Lyapunov spectra in the sub-additive setting, expanding understanding beyond classical additive cases.

Findings

Variational relations between entropy and pressure for level sets of Lyapunov exponents.

Examples demonstrating irregular multifractal behaviors and entropy map discontinuities.

Results applicable without assumptions of unique equilibrium measures or pressure differentiability.

Abstract

For general asymptotically sub-additive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measure-theoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the sub-additive case and in the case that the entropy map that is not upper-semi continuous.