Topological invariance of a strong summability condition in one-dimensional dynamics

Abstract

We say that a rational map satisfies a strong summability condition if, for each critical value v of f belonging to the Julia set, we have for any β>0. We give an equivalent formulation of this property in terms of backward contracting properties of f. We prove that the strong summability condition is a topological invariant for rational maps with one critical point in the Julia set and without parabolic cycles. For unimodal interval maps, we obtain that the strong summability condition is invariant under quasisymmetric conjugacy. © 2012 The Author(s).