Multifractal formalism derived from thermodynamics for general dynamical systems

Vaughn Climenhaga

doi:10.3934/era.2010.17.1

Abstract

We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T$: $ \RR\to \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. Applications include most previously known results, as well as some new ones.

Highlights

A preliminary announcement of the results in this paper is to appear in Electronic Research Announcements.1.1
The basic elements of the multifractal formalism were first proposed by Halsey et al in [HJK+86], where they considered what they referred to as the dimension spectrum or the f (α)-spectrum for dimensions, which characterises an invariant measure μ for a dynamical system f : X → X in terms of the level sets of the pointwise dimension
We obtain our strongest result for the Birkhoff spectrum B(α). This result is given in Theorem 2.1, which applies to continuous maps f : X → X and to functions φ : X → R which lie in a certain class Af ; this class contains, but is not limited to, the space of all continuous functions. For such maps and functions, we show that the function TB : q → P, where P is the pressure, is the Legendre transform of B(α), without any further restrictions on f and φ

Summary

Introduction

A preliminary announcement (without proofs) of the results in this paper is to appear in Electronic Research Announcements. Despite the success of this approach for a number of different classes of systems, there do not appear to be any extant rigorous results which apply to general continuous maps and arbitrary potentials (but see the remark below concerning [FH10]) Such results would give information about the multifractal analysis in settings far beyond those already considered; they would establish the multifractal analysis as a direct corollary of the thermodynamic formalism, rendering Step (3) above automatic, and eliminating the need for the use of a specific toolkit to study the multifractal formalism itself. In order to obtain results on the spectra E(α) and D(α), for which the corresponding local quantities (dμ(x) and hμ(x)) are defined in terms of an invariant measure μ, we need some relationship between μ and a potential function φ This is given by the assumption that μ is a weak Gibbs measure for φ; we observe that there are several cases in which weak Gibbs measures (of one definition or another) are known to exist [Yur[00], Kes[01], FO03, VV08, JR09].

Definitions and results for Birkhoff spectrum

TB is the Legendre transform of the Birkhoff spectrum:

Conformal maps and Lyapunov spectra

Entropy and dimension spectra of weak Gibbs measures

TD is the Legendre transform of the dimension spectrum:

Remarks

Applications

Preparatory results