Multifractal analysis and localized asymptotic behavior for almost additive potentials

TL;DR

This paper develops a multifractal analysis framework for almost-additive potentials on subshifts, introducing new descriptions of the spectrum, including weak concavity and localized asymptotic behavior, with applications to dynamical systems and geometric measure theory.

Contribution

It introduces a novel multifractal analysis approach for almost-additive potentials without bounded distortion, including a new spectrum description and localized asymptotic behavior analysis.

Findings

Expressed Hausdorff spectrum via a conditional variational principle.

Established a new large deviations principle for the potentials.

Computed Hausdorff dimension of localized asymptotic sets.

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 bounded distorsion property assumption. We express the whole Hausdorff spectrum in terms of a conditional variational principle, as well as a new large deviations principle. Our approach provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Another new point is that we consider 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 is naturally related to Birkhoff's ergodic theorem and 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.