Large deviation for weak Gibbs measures and multifractal spectra
The author introduces the class \({\mathcal D}\) of ``medium varying functions'' and corresponding weak Gibbs measures defined on a symbolic shift space. For \(\varphi, \psi \in {\mathcal D}\), \(\psi > 0\) and a weak Gibbs measure \(\mu\) for the potential \(\beta(0)\psi\), he proves that the Helmholtz free energy \(H(t)\) of the stochastic process \((S_{n_R}\varphi, \mu)_{R> 0}\) can be expressed in terms of the topological pressure. The Legendre transform of the convex function \(H\) is called the entropy function. Large deviation estimates are derived using the entropy function. These results are further related to the results of \textit{E. Olivier} [Nonlinearity 12, No. 6, 1571-1585 (1999; Zbl 0955.37003)] on multifractal spectra of Gibbs and \(g\)-measures.
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