Multifractal formalism derived from thermodynamics for general dynamical systems
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.
Received: October 2009; Revised: January 2010; Available Online: April 2010.