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 formKα =\{x∈ X:limn→∞ 1n ∑ᵢ₌₀ⁿ⁻¹ φ( f^i(x))=α \},α∈.For transformations satisfying the specification property we prove the following Variational Principlehₜₒₚ(f,Kα) = ( h_μ(f): μ is invariant and ∫φ \,dμ=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.