Examples of coarse expanding conformal maps

Haı̈ssinsky, Peter; Pilgrim, Kevin M.

doi:10.3934/dcds.2012.32.2403

Cited by 11 publications

(7 citation statements)

References 18 publications

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“…In particular, within this family of fixed degree, there are hyperbolic carpet maps with conformal dimension tending to 1 and to 2. This latter result answers a question of the first author and P. Haïssinsky [HP12]. 1.7.…”

Section: Theorem E We Havesupporting

confidence: 72%

“…Our second alternate proof we present here as a sketch; the motivation comes from [HP12]. Associated to the multi-curve C is a holomorphic virtual endomorphism of spaces π Y , φ Y : Y 1 Ñ Y 0 where Y 0 is a collection of Euclidean annuli of circumference 1 (and geodesic boundary) indexed by the components of C and Y 1 is a collection R j of pairwise disjoint right Euclidean sub-annuli of Y 0 indexed by the components of f ´1pC q homotopic to C. We require that φ Y induces a conformal inclusion Y 1 ãÑ Y 0 and π Y : Y 1 Ñ Y 0 is conformal, a local expanding homothety in the Euclidean coordinates with constant factor 2, and with each component mapping by degree 2.…”

Section: Fat Real Devaney Examplesmentioning

confidence: 99%

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Ahlfors-regular conformal dimension and energies of graph maps

Pilgrim¹,

Thurston²

2021

Preprint

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For a hyperbolic rational map f with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set J f from a family of energies of associated graph maps. Concretely, the dynamics of f is faithfully encoded by a pair of maps π, φ : G 1 Ñ G 0 between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each q ě 1, is a numerical invariant E q rπ, φs, its asymptotic q-conformal energy. We show that the Ahlfors-regular conformal dimension of J f is contained in the interval where E q " 1.Among other applications, we give two families of quartic rational maps with Ahlforsregular conformal dimension approaching 1 and 2, respectively.

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Section: Theorem E We Havesupporting

confidence: 72%

Section: Fat Real Devaney Examplesmentioning

confidence: 99%

Ahlfors-regular conformal dimension and energies of graph maps

Pilgrim¹,

Thurston²

2021

Preprint

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“…From [HP,§3], we have Proposition 1.1. The Ahlfors-regular conformal dimension of X(D) is equal to 1 + λ(D).…”

Section: Annulus Mapsmentioning

confidence: 93%

Quasisymmetrically inequivalent hyperbolic Julia sets

Haı̈ssinsky¹,

Pilgrim²

2012

Rev. Mat. Iberoam.

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Abstract. We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.Quasiconformal geometry is concerned with properties of metric spaces that are preserved under quasisymmetric homeomorphisms. Recall that a homeomorphism h : X → Y between metric spaces is quasisymmetric if there exists a distortion control function η : [0, ∞) → [0, ∞) which is a homeomorphism and which satisfies |h(x) − h(a)|/|h(x) − h(b)| ≤ η(|x − a|/|x − b|) for every triple of distinct points x, a, b ∈ X. We shall say that X and Y are quasisymmetrically equivalent if there exists such a homeomorphism.A basic -even if still widely open-question is to determine whether two given spaces belong to the same quasisymmetry class, once it is known that they are homeomorphic and share the same qualitative geometric properties. This question arises also in the classification of hyperbolic spaces and word hyperbolic groups in the sense of Gromov [BP, Kle, Haï]. Besides spaces modelled on manifolds, very few examples are understood; see nonetheless [Bou] for examples of inequivalent spaces modelled on the universal Menger curve. Here, we focus our attention on compact metric spaces that arise as Julia sets of rational maps. A rational map is hyperbolic if the closure of the set of forward orbits of all its critical points does not meet its Julia set. We address the question of whether the geometry of the Julia set of a hyperbolic rational map is determined by its topology. More precisely, given two hyperbolic rational maps f and g with homeomorphic Julia sets J f and J g , does there exist a quasisymmetric homeomorphism h :Hyperbolic Julia sets serve our purposes for several reasons. First, it rules out elementary local obstructions. For instance, the Julia set of f (z) = z 2 is the Euclidean unit circle S 1 , while that of g(z) = z 2 + 1/4 is a Jordan curve with a cusp at the unique fixed-point, so they are not quasisymmetrically equivalent. Second, if f is hyperbolic, it is locally invertible near J f , and the inverse branches are uniformly Date: July 2, 2018.

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“…• N. Selinger studies compactifications of rational map Teichmüller spaces, [Se]. Work of Bonk,Haïssinsky,Meyer and Pilgrim,[BM,HP1,HP2,HP3,HP4,Me1,Me2,Pi2,Pi3] study postcritically finite branched coverings of S 2 , in particular those with Thurston obstructions. Rivera-Letelier, [Ri], studies some weakly hyperbolic rational maps with the help of the convergence of Thurston's algorithm.…”

Section: Applications Of Thurston's Theoremmentioning

confidence: 99%

Teichmüller spaces and holomorphic dynamics

Buff

Cui

Tan

2014

IRMA Lectures in Mathematics and Theoretical Physics

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One fundamental theorem in the theory of holomorphic dynamics is Thurston's topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston's theorem (the marked Thurston's theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.

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Examples of coarse expanding conformal maps

Cited by 11 publications

References 18 publications

Ahlfors-regular conformal dimension and energies of graph maps

Ahlfors-regular conformal dimension and energies of graph maps

Quasisymmetrically inequivalent hyperbolic Julia sets

Teichmüller spaces and holomorphic dynamics

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