Multifractal formalism derived from thermodynamics

Details Multifractal formalism derived from thermodynamics Multifractal formalism derived from thermodynamics

2010-02-03

arxiv.org/abs/1002.0789v1

Mathematics

Dynamical Systems

52 pages, 3 figures

Scientific paper

We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T\colon \RR\to \RR$ 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. We give applications that include most previously known results, as well as some new ones.

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