Quasisymmetric rigidity in one-dimensional dynamics

Clark, Trevor; van Strien, Sebastian

doi:10.48550/arxiv.1805.09284

Cited by 4 publications

(6 citation statements)

References 34 publications

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“…Fig. 15 The set X in the statement of the QC-Criterion can have a complicated shape. In particular, it is possible that X tiles , as shown on the picture (components of X are shown in yellow).…”

Section: Sketch Of the Proof Of The Qc-rigidity Theoremmentioning

confidence: 99%

“…. c k be a chain in the partial ordering starting from c 0 and consisting of pairwise non-equivalent critical points (we pick a representative in each equivalence class, 15 and the proof below repeats for each such choice).…”

Section: Ergodicity Propertiesmentioning

confidence: 99%

“…Then, meas(X ∩ Comp c 0 F −s (J )) = meas(Comp c 0 F −s (J )). Since 15 A more precise way would be to write…”

Section: Ergodicity Propertiesmentioning

confidence: 99%

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The Dynamics of Complex Box Mappings

et al. 2022

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In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is: To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues. To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics. Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are: the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS; the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009); the QC-Criterion and the Spreading Principle from KSS. The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings. To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings: puzzle pieces shrink to points, (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate. We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattès maps in this setting. We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.

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Section: Sketch Of the Proof Of The Qc-rigidity Theoremmentioning

confidence: 99%

Section: Ergodicity Propertiesmentioning

confidence: 99%

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The Dynamics of Complex Box Mappings

et al. 2022

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“…The notion of asymptotically holomorphic maps goes back at least to [9]. In dynamics, this notion was used in [4,11,12,27,43,61] (see also [22,23] for related material on the more restrictive notion of a uniformly asymptotically conformal (UAC) map).…”

Section: Introductionmentioning

confidence: 99%

Asymptotically holomorphic methods for infinitely renormalizable unimodal maps

Clark¹,

Faria²,

Strien

2022

Ergod. Th. Dynam. Sys.

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The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ( $r>3$ ) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

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“…Each of works mentioned above, except the last one, relies on a so-called Rigidity Theorem in each of their corresponding families. In particular, the proof of density of hyperbolicity in the real polynomial family is based on the following rigidity theorem (see [KSS1,Rigidity Theorem] and [CvS,Theorem 1.1]).…”

Section: Introductionmentioning

confidence: 99%