Rigidity of critical circle mappings II

Faria, Edson de; Melo, Welington de

doi:10.1090/s0894-0347-99-00324-0

Cited by 63 publications

(68 citation statements)

References 17 publications

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“…In this article we take a step towards proving this conjecture for rotation numbers of bounded combinatorial type and critical points of cubic type (s = 3). Further steps are taken in [dFM1] and [dFM2]. Our methods can be adapted to cover all odd exponents s = 2k + 1, k ≥ 1, as well.…”

Section: E De Fariamentioning

confidence: 99%

Asymptotic rigidity of scaling ratios for critical circle mappings

Faria

1999

Ergod. Th. Dynam. Sys.

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Let f be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasi-conformal surgery techniques, we prove that the scaling ratios of f about the critical point are asymptotically independent of f . This settles in particular the golden mean universality conjecture. We introduce the notion of holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of f to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by z → z +θ − 1 2π sin 2πz, θ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for f is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.Mathematics Subject Classification (1991): Primary 58F03, 30F60, 58F23, 32G05.

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Section: E De Fariamentioning

confidence: 99%

Asymptotic rigidity of scaling ratios for critical circle mappings

Faria

1999

Ergod. Th. Dynam. Sys.

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“…The singular points refer either to points where the derivative vanishes (critical points) or where it has a jump discontinuity (break points). In the case of critical circle maps, i.e., circle maps with a single singular point where the derivative vanishes, the first rigidity results were obtained by de Faria and de Melo [4,5]. They established the convergence of renormalizations -the main technical tool in proving rigidity resultsand rigidity for analytic critical circle maps with the same irrational rotation number of bounded type (i.e., with bounded partial quotients) and the same (odd-integer) order of the critical point (i.e., the exponent of the power law behavior of the map in a neighborhood of the critical point).…”

Section: Introductionmentioning

confidence: 95%

C^1-rigidity of circle maps with breaks for almost all rotation numbers

Khanin¹,

Kocić²,

Mazzeo³

2017

Ann. Sci. École Norm. Sup.

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We prove that, for almost all irrational ρ ∈ (0, 1), every two C 2+α-smooth, α ∈ (0, 1), circle diffeomorphisms with a break point, i.e., a singular point where the derivative has a jump discontinuity, with the same rotation number ρ and the same size of the break c ∈ R + \{1}, are C 1-smoothly conjugate to each other. Résumé Nous démontrons que pour presque tous les irrationnels ρ ∈ (0, 1), deux difféomorphismes du cercle C 2+α lisses, α ∈ (0, 1), avec un point de singularité de type rupture où la dérivée a une discontinuité de saut, avec le même nombre de rotation ρ et la même taille de rupture c ∈ R + \{1}, sont C 1-conjugués l'un à l'autre.

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“…The rigidity theory for circle homeomorphisms with singularities and closely related renormalization theory for such maps was a subject of intensive studies in the last 20 years [1,2,3,4,5,6,8,9,19,25,26]. In majority of the papers the following two classes of singularities were studied: critical circle maps, where a map f is smooth but the derivative vanishes at one point f ′ (x cr ) = 0, and circle maps with breaks, where a map f is smooth outside of one point x br and it has a jump discontinuity of the first derivative at x br .…”

Section: Introductionmentioning

confidence: 99%

“…For critical maps, it is conjectured, and in many cases proved, that Diophantine conditions for C 1 rigidity are not needed. For a more restrictive class of rotation numbers, which includes irrational numbers of bounded type, one can prove that the conjugacy is C 1+ǫ smooth for some ǫ > 0, see [5,6]. We should add that all the rigidity results in the critical case are proved only for γ being an odd integer number greater than 1, although it is generally believed that the rigidity statements remain valid for all γ > 1.…”

Section: Introductionmentioning

confidence: 99%

Circle homeomorphisms with breaks with no $C^{2-ν}$ conjugacy

Goncharuk¹,

Khanin²,

Kudryashov³

2021

Preprint

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The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved [14,15,18,20] that under mild conditions of the Diophantine type on the rotation number any two C 2+α smooth circle homeomorphisms with a break point are C 1 smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be C 2−ν even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from the linearizable case of circle diffeomorphisms where conjugacy is arbitrarily smooth, or even analytic, for sufficiently smooth diffeomorphisms.

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Rigidity of critical circle mappings II

Cited by 63 publications

References 17 publications

Asymptotic rigidity of scaling ratios for critical circle mappings

Asymptotic rigidity of scaling ratios for critical circle mappings

C^1-rigidity of circle maps with breaks for almost all rotation numbers

Circle homeomorphisms with breaks with no $C^{2-ν}$ conjugacy

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