Ergodic Theory and Dynamical Systems

A connecting lemma for rational maps satisfying a no-growth condition

a1 Departamento de Matemáticas, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile (e-mail:

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rivera-letelier j   [Google Scholar] 


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 one. 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 is sets having a Markov property for backward iterates that are holomorphic analogues of nice intervals in real one-dimensional dynamics.

(Published Online February 12 2007)
(Received December 29 2005)
(Revised July 13 2006)