Multifractal Analysis for Quotients of Birkhoff Sums for Countable Markov Maps

This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville–Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions are also given.