Marc Kesseb{\"o}hmer
Large deviation for weak Gibbs measures and multifractal spectra
Preprint series: Mathematica Gottingensis

MSC:

28A80 Fractals

60F10 Large deviations

Abstract: We introduce the class of `medium varying functions' and corresponding
weak Gibbs measures both defined on a symbolic shift space. We prove that the
free Helmholtz energy of the stochastic process of a randomly stopped
Birkhoff sum measured by a weak Gibbs measure can be expressed in terms of the
topological pressure. This leads to the notion of the multifractal entropy function
which provides large deviation bounds. The multifractal entropy function
can be considered as a generalization of the multifractal spectrum as
they coincide (up to constants) when for instance Gibbs or \( g \)--measures are
involved.
Keywords: large deviation, multifracals, weak Gibbs measures, Birkhoff sum, symbolic shift space, transfer operator