Weighted thermodynamic formalism and applications

TL;DR

This paper develops a weighted thermodynamic formalism for subshifts, establishing variational principles, uniqueness, and Gibbs properties of equilibrium states, with applications to multifractal analysis of Birkhoff averages and self-affine measures.

Contribution

It introduces weighted topological pressure and equilibrium states for asymptotically sub-additive potentials, extending thermodynamic formalism to higher dimensions and applications.

Findings

Proved uniqueness and Gibbs property of weighted equilibrium states for full shifts.

Established a weighted variational principle for asymptotically sub-additive potentials.

Applied the theory to multifractal analysis of Birkhoff averages and self-affine symbolic spaces.

Abstract

Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $\Phi$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $\Phi$. We setup the weighted variational principle. In the case that $X, Y$ are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional weighted thermodynamic formalism. As an application, we conduct the multifractal analysis for a new type of level sets associated with Birkhoff averages, as well as for weak Gibbs measures associated with asymptotically additive potentials on self-affine symbolic spaces.