A Multifractal Analysis of Gibbs Measures for Conformal Expanding Maps and Markov Moran Geometric Constructions
نویسنده
چکیده
We establish the complete multifractal formalism for Gibbs measures for confor-mal expanding maps and Markov Moran geometric constructions. Examples include Markov maps of an interval, hyperbolic Julia sets, and conformal toral endomorphisms. This paper describes the multifractal analysis of measures invariant under dynamical systems. The concept of a multifractal analysis was suggested by several physicists in the seminal paper HJKPS] and became a popular interdisciplinary subject of study. A search of several electronic databases showed that there are now hundreds of related papers in the physical and mathematical literature. The rst rigorous multifractal analysis was carried out in CLP] for a special class of measures invariant under some one-dimensional Markov maps, and in Ra] for Gibbs measures for Cookie-Cutter maps. Lopes Lo] studied the measure of maximal entropy for a hyperbolic Julia set. Recently, Simpelaere Si] eeected a complete multifractal analysis for Gibbs measures of Axiom A surface diieomorphisms. The two major components of the multifractal analysis are the Hentschel-Procaccia (HP) spectrum for dimensions and the f() spectrum for dimensions. We will provide motivation for introducing these spectra by considering the BRS (Bowen-Ruelle-Sinai) measures on a hyperbolic attractors. a compact hyperbolic attractor for g. For simplicity, we assume that g is topologically mixing on. In B], Bowen showed that the evolution of the Lebesgue measure in a
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