Multifractal analysis of geodesic flows on surfaces without focal points

Abstract

In this paper, we study multifractal spectra of the geodesic flows on compact rank 1 surfaces without focal points. We compute the entropy of the level sets for the Lyapunov exponents and establish a lower bound for their Hausdorff dimension in terms of the pressure function and its Legendre transform. In doing so, we employ and generalize results of Burns and Gelfert for non-positively curved surfaces and construct an increasingly nested sequence of basic sets in the complement of the singular set on which the geodesic flow is non-uniformly hyperbolic. Such a sequence of basic sets eventually contains any given basic set.