Higher-dimensional Multifractal Analysis for the Cusp Winding Process on Hyperbolic Surfaces

We perform a multifractal analysis of the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with m≥1 cusps. Our main theorem establishes a conditional variational principle for the Hausdorff dimension spectrum of the multi-cusp winding process. Moreover, we show that the dimension spectrum defined on (0,∞)m is real analytic. To prove the main theorem, we use a countable Markov shift with a finitely primitive transition matrix and a thermodynamic formalism.