Abstract
We introduce the class of `medium varying functions' and correspondingweak Gibbs measures both defined on a symbolic shift space. We provethat the Helmholtz free energy of the stochastic process of a randomlystopped Birkhoff sum measured by a weak Gibbs measure can be expressedin terms of the topological pressure. This leads to the notion of themultifractal entropy function which provides large deviation bounds.Weshow that the multifractal entropy functions coincide (up toconstants)with multifractal spectra when Gibbs or g-measures are involved.
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