The local Hölder exponent for the entropy of real unimodal maps

Abstract

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ (equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Holder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Holder exponent equals exactly, up to a factor log 2, the value of the function at that point. This confirms a conjecture of Isola and Politi (1990), and extends a similar result for the dimension of invariant subsets of the circle.