Hyperbolicity of renormalization of critical circle maps

Abstract

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...

Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks

“…Despite serious efforts, in the case of critical circle maps this conjecture is still open. In the analytic category, an analogous claim is indeed true, as was proved by de Faria and de Melo [9,10], for bounded-type irrational rotation numbers, and extended to all irrational rotation numbers by Yampolsky [31]. The type of singularity in the case of critical circle maps is characterized by the order of the critical point.…”

Complex Bounds for Real Maps

“…For critical circle maps, there is a renormalization theory that is closely related to the theory for unimodal maps, see for example [Ya2,KT,dFdM1,dFdM2].…”

Renormalization of vector fields

“…The main drawback with this approach is that commuting maps do not constitute a manifold, which makes it hard to discuss some important aspects of renormalization, such as the hyperbolicity of the renormalization operator. (Similar problems are also encountered in one-dimensional dynamics, which lead to the development of alternative approaches, such as the "cylinder renormalization" for Siegel disks [125,56]. )…”