MULTIFRACTAL FORMALISM DERIVED FROM THERMODYNAMICS FOR GENERAL DYNAMICAL SYSTEMS

We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions T : R -> R 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.