The maximal entropy measure detects non-uniform hyperbolicity

Rivera-Letelier, Juan

doi:10.4310/mrl.2010.v17.n5.a5

Cited by 10 publications

(9 citation statements)

References 21 publications

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“…The proof of Theorem 2 is based on a global inductive estimate of conformal densities distributed over elements of Yoccoz partitions. The main technical difficulty to overcome is a tendency of conformal measures to excessively concentrate around recurrent critical points [18,45].…”

Section: Deep Pointsmentioning

confidence: 99%

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Analytic structures and harmonic measure at bifurcation locus

Graczyk¹,

Świątek²

2019

Preprint

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We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci M d for unicritical polynomials f c (z) = z d +c. It is known that these parameters are structurally unstable and have stochastic dynamics. We prove C 1+ α d − -conformality, α = 2 − HD (J c 0 ), of the parameter-phase space similarity maps Υ c 0 (z) : C → C at typical c 0 ∈ ∂M d and establish that globally quasiconformal similarity maps Υ c 0 (z), c 0 ∈ ∂M d , are C 1 -conformal along external rays landing at c 0 in C \ J c 0 mapping onto the corresponding rays of M d . This conformal equivalence leads to the proof that the z-derivative of the similarity map Υ c 0 (z) at typical c 0 ∈ ∂M d is equal to 1/T (c 0 ), whereis the transversality function. The paper builds analytical tools for a further study of the extremal properties of the harmonic measure on ∂M d , [25]. In particular, we will explain how a nonlinear dynamics creates abundance of hedgehog neighborhoods in ∂M d effectively blocking a good access of ∂M d from the outside.

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Section: Deep Pointsmentioning

confidence: 99%

“…Even though not explicitely stated, the concept of hedgehog layers was introduced by J. Riviera-Letelier in his study of porosity at critical recurrent points for rational functions, see the proof of Theorem C' in ( [45]).…”

Section: Geometric Applicationsmentioning

confidence: 99%

Analytic structures and harmonic measure at bifurcation locus

Graczyk¹,

Świątek²

2019

Preprint

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“…Remark 3.3. The TCE property is detected by the maximal entropy measure: indeed it is equivalent to the property that the measure of small balls satisfies an estimate of the form µpBpx, rqq Á r θ for some θ ą 0 and for every x P J (see [36]). It is not difficult to see that if such an estimate holds for every x P U , where U is any open set interesting J f , then it holds everywhere (possibly with a different θ).…”

Section: Local Contractions and Preperiodic Pointsmentioning

confidence: 99%

When do two rational functions have locally biholomorphic Julia sets?

Dujardin¹,

Favre²,

Gauthier³

2022

Preprint

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In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given onedimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.

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“…Lemma 9 (Lemma 1, [RL10]). Let (X, dist) be a compact metric space and let µ be a doubling measure on X.…”

Section: Doubling Implies Non-recurrent Critical Pointsmentioning

confidence: 99%

“…Acknowledgments. The main idea of this paper came to the author after several discussions with Juan Rivera-Letelier on his work [RL10], and he read an earlier version very carefully. I am very grateful to him for those stimulating conversations and for his useful comments and corrections to an earlier version of this paper.…”

Section: Introductionmentioning

confidence: 99%

Interval maps quasi-symmetrically conjugate to a piecewise affine map

2014

Journal of Mathematical Analysis and Applications

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Consider a multimodal interval map f of C 3 with non-flat critical points. We establish several characterizations of the map f is quasisymmetrically conjugated to a piecewise affine map in the case f is topologically exact and all of its periodic points are hyperbolic repelling. In particular, we give a negative answer to a question posted by Henk Bruin in [Bru96].2010 Mathematics Subject Classification. 37E05.

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The maximal entropy measure detects non-uniform hyperbolicity

Cited by 10 publications

References 21 publications

Analytic structures and harmonic measure at bifurcation locus

Analytic structures and harmonic measure at bifurcation locus

When do two rational functions have locally biholomorphic Julia sets?

Interval maps quasi-symmetrically conjugate to a piecewise affine map

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