American Institute of Mathematical Sciences

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  2010, 17: 1-11. doi: 10.3934/era.2010.17.1

Multifractal formalism derived from thermodynamics for general dynamical systems

Vaughn Climenhaga 1,

1. 

Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802

Received  October 2009 Revised  January 2010 Published  April 2010

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.
Keywords: Birkhoff spectrum, dimension spectrum., thermodynamic formalism, topological pressure, Multifractal analysis.
Mathematics Subject Classification: 37C4.
Citation: Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1
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