Mean of the singularities of a Gibbs measure (Q1924053)

Suppose \(g\) is a (piecewise) \(C^{1 + \gamma}\) expanding Markov transformation of the circle (the unit interval). Let \(\mu\) be a \(g\)-invariant measure, which, in addition, is the Gibbs measure associated with a real-valued \(\gamma\)-Hölder function \(\varphi\). It is shown that, for any \(\beta\in\mathbb{R}\), the following limit exists and satisfies \[ M (\beta) = \lim_{r\to 0} {\log \int\mu \bigl(B(x,r)\bigr)^\beta d\mu(x) \over \log r}= \inf \left\{{h_\rho + (\beta+1) \int\varphi d \rho- (\beta+1) P_\varphi \over \int-\log |g'|d\rho} \right\}, \] where the infimum is taken over all \(g\)-invariant measures \(\rho\). Here \(B(x,r)\) is the ball of radius \(r\) around \(x\), \(h_\rho\) is the entropy of \((g,\rho)\), and \(P_\varphi = \sup\{h_\rho + \int\varphi d \rho\}\) is the pressure of \(\varphi\). The function \(\beta\mapsto M (\beta)\) is shown to be real analytic, strictly increasing, concave, and strictly concave unless \(\varphi\) and \(-\log |g' |\) are \(C^\gamma\)-homologous. This is then extended to Axiom A \(C^2\) diffeomorphisms on compact two-dimensional manifolds, essentially by using one-dimensional analysis on the stable and unstable manifolds, respectively.