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 on X we consider sets of the formK_ =\x X: ₍ 1n ₈=₀^{n-1 (fⁱ (x) ) = \},. For transformations satisfying the specification property we prove the following Variational Principleh ₓ₎₏ (f, K_) = (h_ (f): is invariant and \, d=), where h ₓ₎₏ (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 multifractal spectra and establish a contraction principle.