The Collet–Eckmann condition for rational functions on the Riemann sphere
TLDR
In this paper, it was shown that the set of Collet-Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.Abstract:
We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.read more
Citations
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Journal ArticleDOI
Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
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A connecting lemma for rational maps satisfying a no-growth condition
TL;DR: In this paper, the authors introduce a non-uniform hyperbolicity condition for complex rational maps that does not involve a growth condition and show that this condition is weaker than the Collet?Eckmann condition, and than the summability condition with exponent one.
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Large deviation principles for non-uniformly hyperbolic rational maps
TL;DR: In this paper, the authors show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called Topological Collet-Eckmann.
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Asymptotic expansion of smooth interval maps
TL;DR: In this article, it was shown that several natural notions of nonuniform hyperbolicity coincide under topological conjugacy with respect to the Collet-Eckmann condition for unicritical maps.
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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.
References
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Book
Real and complex analysis
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Book
Principles of Algebraic Geometry
Phillip Griffiths,Joe Harris +1 more
TL;DR: In this paper, a comprehensive, self-contained treatment of complex manifold theory is presented, focusing on results applicable to projective varieties, and including discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex.
Journal ArticleDOI
The Dynamics of the Henon Map
TL;DR: In this paper, the Henon map with expansion combined with strong contraction is modeled on the treatment of the one-dimensional system x→1-ax 2 and the perturbation of a from the value a=2 and b small.
Book
Plane algebraic curves
TL;DR: The projective closure of algebraic curves and their equations are discussed in this article, along with a discussion of the implicit function theorem and the Harnack inequality of singularities.
Journal ArticleDOI
Absolutely continuous invariant measures for one-parameter families of one-dimensional maps
TL;DR: In this paper, the set of parameter values λ for which λ has an invariant measure absolutely continuous with respect to Lebesgue measure has been studied and shown to have positive measure for two classes of maps.