Density of hyperbolicity for classes of real transcendental entire functions and circle maps

Rempe, Lasse; Strien, Sebastian van

doi:10.1215/00127094-2885764

Cited by 18 publications

(16 citation statements)

References 46 publications

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“…For the unimodal case see [GS2], [GS3], [Ly2], [Ko], and for the general (real) multimodal case see [KSvS1] and [KSvS2]. Density of hyperbolicity is also known within certain spaces of real transcendental maps and in the Arnol'd family, see [RvS1] and [RvS2].…”

Section: Previous Resultsmentioning

confidence: 99%

Quasisymmetric rigidity in one-dimensional dynamics

Clark,

van Strien

2018

Preprint

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In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary assumptions are satisfied). The aim of this paper is to conclude this programme in a natural class of C 3 mappings. Examples of such rigidity were established previously, but not, for example, for real polynomials with non-real critical points.Our results are also new for analytic mappings. The main new ingredients of the proof in the real analytic case are (i) the existence of infinitely many (complex) domains associated to its complex analytic extension so that these domains and their ranges are compatible, (ii) a methodology for showing that combinatorially equivalent complex box mappings are qc conjugate, (iii) a methodology for constructing qc conjugacies in the presence of parabolic periodic points.For a C 3 mapping, the dilatation of a high iterate of any complex extension of the real map will in general be unbounded. To deal with this, we introduce dynamically defined qc\bg partitions, where the appropriate mapping has bounded quasiconformal dilatation, except on sets with 'bounded geometry'. To obtain such a partition we prove that we have very good geometric control for infinitely many dynamically defined domains. Some of these results are new even for real polynomials, and in fact an important sequence of domains turn out to be quasidiscs. This technology also gives a new method for dealing with the infinitely renormalizable case.We will briefly also discuss why quasisymmetric rigidity is such a useful property in one-dimensional dynamics.

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Section: Previous Resultsmentioning

confidence: 99%

Quasisymmetric rigidity in one-dimensional dynamics

Clark,

van Strien

2018

Preprint

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“…, ∞, ω. One can also show that one has density of hyperbolicity for real transcendental maps and within full families of real analytic maps [RvS,CvS2]. One can apply the techniques of [KSS1] to prove complex bounds and qc rigidity for real analytic mappings and even in some sense for smooth interval maps, see [CvST, CvS].…”

Section: Pathologies Of General Complex Box Mappings There Existsmentioning

confidence: 99%

The dynamics of complex box mappings

Clark,

Drach,

Kozlovski

et al. 2021

Preprint

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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 [KSS1]; -the Covering Lemma (which controls the moduli of pullbacks of annuli) from [KL1]; -the QC-Criterion and the Spreading Principle from [KSS1].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 [KvS, KSS1] 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.Date: May 19, 2021.Acknowledgments. The research of the second author was partially supported by the advanced grant 695 621 "HOLO-GRAM" of the European Research Council (ERC), which is gratefully acknowledged. We would also like to thank Dzmitry Dudko and Dierk Schleicher for many stimulating discussions and encouragement during our work on this project, Weixiao Shen and Mikhail Hlushchanka for helpful comments.

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“…Quasisymmetric rigidity can also be proved for a large class of real transcendental maps, see [RvS1] and [RvS2]. Another motivation is to extend results about monotonicity of entropy for real polynomials with only real critical points, see [BvS], to real polynomials with non-real critical points.…”

Section: Then the Conjugacy Between F And G Is Quasisymmetricmentioning

confidence: 99%

Complex Bounds for Real Maps

Clark

Strien

Trejo

2017

Commun. Math. Phys.

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Abstract:In this paper we prove complex bounds, also referred to as a priori bounds for C 3 , and, in particular, for analytic maps of the interval. Any C 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval. We associate to such a map, a complex box mapping, which provides a kind of Markov structure for the dynamics. Moreover, we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain. Such bounds show that the first return maps to these domains are well controlled, and consequently such bounds form one of the corner stones in many recent results in one-dimensional dynamics, for example: renormalization theory, rigidity, density of hyperbolicity, and local connectivity of Julia sets.

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Density of hyperbolicity for classes of real transcendental entire functions and circle maps

Cited by 18 publications

References 46 publications

Quasisymmetric rigidity in one-dimensional dynamics

Quasisymmetric rigidity in one-dimensional dynamics

The dynamics of complex box mappings

Complex Bounds for Real Maps

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