A connecting lemma for rational maps satisfying a no growth condition

Abstract

We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann condition, and than the summability condition with exponent~1. Our main result is a connecting lemma for Backward Contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rigidity properties of quasi-conformal maps. We also prove that the Lebesgue measure of the Julia set of a Backward Contracting rational map is zero, when it is not the whole Riemann sphere. The basic tool of this article are sets having a Markov property for backward iterates, that are holomorphic analogues of nice intervals in real one-dimensional dynamics.