Multifractal analysis is applied to the Gauss dynamical system associated to continued fractions.

First, by applying the Ruelle operator theory, the complete multifractal spectra of the Khinchine exponents and Lyapunov exponents of continued fractions are given. Then it is showed that the Hausdorff dimensions of the sets of numbers with prescribed frequencies of partial quotients in their continued fraction expansions can be given by a modified variational principle. Last, some sets of extremely non-mormal continued fractions are proven to be of Hausdorff dimension one-half. The results on Gauss dynamics can be generalized to interval maps of infinitely many branches.