Localized asymptotic behavior for almost additive potentials

TL;DR

This paper analyzes the multifractal structure of almost additive potentials on subshifts, revealing a new weakly concave spectrum description and computing Hausdorff dimensions of localized asymptotic sets.

Contribution

It introduces a novel approach to multifractal analysis without regularity assumptions, providing new insights into the spectrum's structure and localized asymptotic behavior.

Findings

Spectrum described as weakly concave

Hausdorff dimension of localized sets computed

Applications to dynamical systems and harmonic measure

Abstract

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} \phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\R^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.