Large deviation for weak Gibbs measures and multifractal spectra

We introduce the class of 'medium varying functions' and corresponding weak Gibbs measures both defined on a symbolic shift space. We prove that the Helmholtz free 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. We show that the multifractal entropy functions coincide (up to constants) with multifractal spectra when Gibbs or g-measures are involved.