On the variational principle for the topological entropy of certain non-compact sets

For a continuous transformation f of a compact metric space (X, d) and any continuous function psi on X we consider sets of the form [GRAPHICS] For transformations satisfying the specification property we prove the following Variational Principle [GRAPHICS] where h(top)(f,) is the topological entropy of non-compact sets. Using this result we are able to obtain a complete description of the multifractal spectrum for Lyapunov exponents of the so-called Manneville-Pomeau map, which is an interval map with an indifferent fixed point. We also consider multi-dimensional multiftactal spectra and establish a contraction principle.