Measure-theoretic degrees and topological pressure for non-expanding transformations

Abstract

We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree off|Λ:Λ→Λ, with respect to the measure μϕ on the fractal set Λ. In our case, the equilibrium measure μϕ is the unique linear functional in C(Λ)⁎ tangent to the pressure function P(⋅):C(Λ)→R at ϕ. In particular, for the measure of maximal entropy μ0 of f|Λ, we obtain the asymptotic degree of f|Λ, which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n→∞; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μϕ, with respect to arbitrary iterates fm, m≥2. We then show that a formula for the topological pressure P(ϕ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μϕ-measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets.