Julia and John revisited

Mihalache, N

doi:10.4064/fm215-1-4

Cited by 13 publications

(15 citation statements)

References 15 publications

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“…Each doubling measure satisfies the property described in Theorem B, see Lemma 1. An analogous upper bound holds for the maximal entropy measure of each rational map f : there are C > 0, α > 0 such that for all x ∈ J(f ) and r > 0 we have [16] that each Fatou component of a semi-hyperbolic rational map with connected Julia set is a John domain with a uniform constant.…”

Section: Introductionmentioning

confidence: 93%

“…One of the implications of this theorem is given by an extension of one of the results of Carleson, Jones and Yoccoz, shown by Mihalache in [16]: each Fatou component of a semi-hyperbolic rational map is a John domain, see also [27] for the case of connected Julia sets. See §1.2 for several examples of rational maps showing that the converse of this last result does not hold in general.…”

Section: Introductionmentioning

confidence: 96%

“…Another interesting example, pointed out in [16], is given by the mating of quadratic polynomials with a Siegel disk, that was studied by Yampolsky and Zakeri in [25]. In fact, this rational map is not semi-hyperbolic as it has a Siegel disk and yet each of its Fatou components is a quasi-disk and hence a John domain.…”

Section: Introductionmentioning

confidence: 99%

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The maximal entropy measure detects non-uniform hyperbolicity

Rivera-Letelier

2010

Mathematical Research Letters

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Abstract. We characterize two of the most studied non-uniform hyperbolicity conditions for rational maps, semi-hyperbolicity and the topological Collet-Eckmann condition, in terms of the maximal entropy measure.With the same tools we give an extension of the result of Carleson, Jones and Yoccoz that semi-hyperbolicity characterizes those polynomial maps whose basin of attraction of infinity is a John domain, to rational maps having a completely invariant attracting basin.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 96%

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The maximal entropy measure detects non-uniform hyperbolicity

Rivera-Letelier

2010

Mathematical Research Letters

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“…The paper [10] deals with polynomials p. The bounded turning property for rational functions can be found in [24,Corollary 2], (see the remark regarding [24, Corollary 2 in §4]). For a proof of (6) in the rational case, see the proof of Theorem A in [26] (using the notation of that proof, notice that G(x, δ, deg( p), p) = N, use that the sequence r n = diam Comp x p −n (B( p n (x), δ)) satisfies r n ≤ ξ n for some ξ < 1 by [26, Proposition 2.5], and notice that r n+1 /r n is bounded from below; it follows that for every sufficiently small r there is n with r n comparable to r ).…”

Section: Quasisymmetry Of the Conjugacymentioning

confidence: 99%

Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Eroglu¹,

Rohde²,

Solomyak³

2009

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385709000789How to cite this article: KEMAL ILGAR EROĞLU, STEFFEN ROHDE and BORIS SOLOMYAK (2010). Quasisymmetric conjugacy between quadratic dynamics and iterated function systems.Abstract. We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the 'overlap set' O is finite, and which are 'invertible' on the attractor A, in the sense that there is a continuous surjection q : A → A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at O. We suppose also that there is a rational function p with the Julia set J such that (A, q) and (J, p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz, λz + 1} where λ is a complex parameter in the unit disk, such that its attractor A λ is a dendrite, which happens whenever O is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map q λ on A λ . If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map p c (z) = z 2 + c, with the Julia set J c such that (A λ , q λ ) and (J c , p c ) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

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“…Finally, we will prove that condition (4) of homogeneity is satisfied. This follows easily from the fact that the Fatou components of a semi-hyperbolic rational map are uniform John domains in the spherical metric [Mih,Proposition 9]. Since we are only interested in the bounded Fatou components, we can use instead the Euclidean metric.…”

Section: Homogeneous Sets and Julia Setsmentioning

confidence: 99%

Semi-hyperbolic rational maps and size of Fatou components

Ntalampekos¹

2018

Ann. Acad. Sci. Fenn. Math.

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Recently, Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpiński carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpiński carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.

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Julia and John revisited

Cited by 13 publications

References 15 publications

The maximal entropy measure detects non-uniform hyperbolicity

The maximal entropy measure detects non-uniform hyperbolicity

Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

Semi-hyperbolic rational maps and size of Fatou components

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