A connecting lemma for rational maps satisfying a no-growth condition
By Juan Rivera-Letelier. Ergodic Theory and Dynamical Systems, 2006
48 citations recorded by the source corpus.
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.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/a4be803a423095ec6f2ae1c6012b5f9319e1aaca
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Citations in the corpus
- Statistical properties of topological Collet–Eckmann maps (2006, methodology): “The construction of the induced map is based on the concept of “nice couple” introduced in [40], which is closely related to nice intervals of real one-dimensional dynamics (§§4.”
- Large derivatives, backward contraction and invariant densities for interval maps (2008, methodology): “• for each c ∈ Crit(f), there exists a number lc > 1 (called the order of c) and C3 diffeomorphisms φ, ψ of R with φ(c) = ψ(f(c)) = 0 such that”
- Nice Inducing Schemes and the Thermodynamics of Rational Maps (2008, methodology): “The following proposition (which we owe to Shen), though not used later on, sheds some light on the definitions above; compare with the construction of nice couples in §A.1 (Theorem D), and in [RL07, §6].”