Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Multifractal formalism derived from thermodynamics for general dynamical systems

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

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Vaughn Climenhaga - Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, United States (email)

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.

Keywords:  Multifractal analysis, thermodynamic formalism, topological pressure, Birkhoff spectrum, dimension spectrum.
Mathematics Subject Classification:  37C45.

Received: October 2009;      Revised: January 2010;      Available Online: April 2010.