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 φ on X we consider sets of the form Kα = {x ∈ X : lim n→∞ 1/n n−1Σi=0 φ(f^i(x)) = α}, α ∈ R. For transformations satisfying the specification property we prove the following Variational Principle htop(f, Kα) = sup(hµ(f): µ is invariant and ∫φdµ = α), where htop(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.