Geometric rigidity for class $\mathcal {S}$ of transcendental meromorphic functions whose Julia sets are Jordan curves

Urbański, Mariusz

doi:10.1090/s0002-9939-09-09918-3

Cited by 3 publications

(3 citation statements)

References 13 publications

(25 reference statements)

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“…Julia sets of hyperbolic rational maps f are geometrically rigid in the sense that if J (f ) is a Jordan curve, then either J (f ) is a circle in Ĉ or dim H J (f ) > 1, where dim H stands for Hausdorff dimension. (More recently, Urbański [16] has extended this result to a class of meromorphic functions with finitely many singularities). In this example, we cannot deform the circle without transforming it into a fractal.…”

Section: Introductionmentioning

confidence: 92%

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Dynamics of hyperbolic correspondences

Siqueira

2021

Ergod. Th. Dynam. Sys.

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This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

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Section: Introductionmentioning

confidence: 92%

“…In any case, the sequence (z i ) is contained in D c . Since we have already proved that (16) holds, it follows that every point of the sequence (z i ) is in J c . THEOREM 5.5.…”

Section: Stability and Hyperbolicitymentioning

confidence: 99%

Dynamics of hyperbolic correspondences

Siqueira

2021

Ergod. Th. Dynam. Sys.

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“…We show that the path-space measure P is quasi-invariant, and we compute the corresponding Radon-Nikodym derivative. Our motivation derives from the need to realize multiresolution models in in a general setting of dynamical systems as they arise in a host of applications: in symbolic dynamics, e.g., [BJO04,BJKR02], in generalized multiresolution model, e.g., [DJ09]; in dynamics arising from an iteration of substitutions, e.g., [Bea91]; in geometric measure theory, and for Iterated Function Systems (IFS), e.g., [Hut81,Urb09]; or in stochastic analysis, e.g., [AJ12, Hut81, MNB16, GF16, ZXL16, YPL16, TSI + 15, Pes13, KLTMV12].…”

Section: Introductionmentioning

confidence: 99%

Transfer Operators, Induced Probability Spaces, and Random Walk Models

Jørgensen¹,

Tian²

2015

Preprint

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We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator R subject to a set of axioms, and a given endomorphism in a compact Hausdorff space X. Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair (h, λ), where h is an R-harmonic function on X, and λ is a given positive measure on X subject to a certain invariance condition defined from R. With this we show that there are then discrete-time random-walk realizations in explicit path-space models; each associated to a probability measures P on path-space, in such a way that the initial data allows for spectral characterization: The initial endomorphism in X lifts to an automorphism in path-space with the probability measure P quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multi-resolutions in L 2 of P in the sense of Lax-Phillips scattering theory.

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Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

Mayer¹,

Rempe²

2011

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385711000332How to cite this article: VOLKER MAYER and LASSE REMPE (2012). Rigidity and absence of line elds for meromorphic and Ahlfors islands maps.Abstract. In this paper, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational and transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Epstein. In fact, we prove a more general result on the absence of invariant differentials, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for log | f |, which are relevant to a number of well-known rigidity questions. In particular, we prove the absence of continuous line fields on the Julia set of any transcendental entire function.

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Geometric rigidity for class $\mathcal {S}$ of transcendental meromorphic functions whose Julia sets are Jordan curves

Cited by 3 publications

References 13 publications

Dynamics of hyperbolic correspondences

Dynamics of hyperbolic correspondences

Transfer Operators, Induced Probability Spaces, and Random Walk Models

Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

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