The Attractor of Renormalization¶and Rigidity of Towers of Critical Circle Maps

Yampolsky, Michael

doi:10.1007/pl00005561

Cited by 48 publications

(37 citation statements)

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“…The space of critical commuting pairs modulo affine conjugacy will be denoted by C; its subset consisting of pairs ζ with χ(ζ) = ∞ will be denoted by S ∞ . Renormalization is a transformation R : C \ C ∞ → C. It is not difficult to show (see [Ya2]) that this transformation is injective:…”

Section: Explanation Of Universality: Lanford's Programmentioning

confidence: 99%

“…To circumvent this obstacle, in this paper we introduce a different renormalization construction, a renormalization of critical cylinder maps. This construction is strongly motivated by a so-called parabolic renormalization, a degenerate case of renormalization, which we studied in detail in [Ya2]. There is a natural functorial relation between the renormalization of cylinder maps and the usual one which allows us to transfer the results back and forth.…”

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confidence: 98%

“…Later de Faria and de Melo [dFdM2] used McMullen's work to show that the convergence to the horseshoe is geometric. The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound.Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference. Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case.…”

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confidence: 99%

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confidence: 99%

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Hyperbolicity of renormalization of critical circle maps

Yampolsky¹

2003

Publ. math., Inst. Hautes Étud. Sci.

139

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The renormalization theory of critical circle maps was developed in the late 1970's-early 1980's to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, and are analogous to Feigenbaum universality in the dynamics of unimodal maps. In the works of Ostlund et al. [ORSS] and Feigenbaum et al. [FKS] they were translated into hyperbolicity of a renormalization transformation.The first renormalization transformation in one-dimensional dynamics was constructed by Feigenbaum and independently by Coullet and Tresser in the setting of unimodal maps. The recent spectacular progress in the unimodal renormalization theory began with the seminal work of Sullivan [Sul1,Sul2,MvS]. He introduced methods of holomorphic dynamics and Teichmüller theory into the subject, developed a quadratic-like renormalization theory, and demonstrated that renormalizations of unimodal maps of bounded combinatorial type converge to a horseshoe attractor. Subsequently, McMullen [McM2] used a different method to prove a stronger version of this result, establishing, in particular, that renormalizations converge to the attractor at a geometric rate. And finally, Lyubich [Lyu4,Lyu5] constructed the horseshoe for unbounded combinatorial types, and showed that it is uniformly hyperbolic, with one-dimensional unstable direction, thereby bringing the unimodal theory to a completion.The renormalization theory of circle maps has developed alongside the unimodal theory. The work of Sullivan was adapted to the subject by de Faria, who constructed in [dF1,dF2] the renormalization horseshoe for critical circle maps of bounded type. Later de Faria and de Melo [dFdM2] used McMullen's work to show that the convergence to the horseshoe is geometric. The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound.Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference. Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case. The first part of Lyubich's work was to to endow the This work was partially supported by NSF Grant DMS-9804606.2 M I C H A E L Y A M P O L S K Y ambient space of the renormalization transformation with the structure of a complexanalytic manifold, with respect to which renormalization is an analytic operator. He then showed that the stable sets of periodic points of this operator are codimension one submanifolds and used an argument based on McMullen's Tower Rigidity Theorem and an infinite dimensional version of the Schwarz Lemma to show that renormalization is a strict contraction in the stable direction. The second part of Lyubich's work was to show the existence of an unstab...

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Section: Explanation Of Universality: Lanford's Programmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hyperbolicity of renormalization of critical circle maps

Yampolsky¹

2003

Publ. math., Inst. Hautes Étud. Sci.

139

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“…It has been particularly successful in one-dimensional dynamics, explaining universality for unimodal maps, critical circle maps and maps with a Siegel disk (see e.g. [1,4,12,15,18,19,21,22] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps

Lilja

2017

Commun. Math. Phys.

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Since its inception in the 1970s at the hands of Feigenbaum and, independently, Coullet and Tresser the study of renormalization operators in dynamics has been very successful at explaining universality phenomena observed in certain families of dynamical systems. The first proof of existence of a hyperbolic fixed point for renormalization of area-preserving maps was given by Eckmann et al. (Mem Am Math Soc 47(289):vi+122, 1984). However, there are still many things that are unknown in this setting, in particular regarding the invariant Cantor sets of infinitely renormalizable maps. In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the EckmannKoch-Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of smooth curves requires new techniques.

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