  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.
