Gabriela Estevez, Instituto de Matematica e Estatıstica, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, 24.210-201, Bloco H, Campus do Gragoata, Niteroi, Rio de Janeiro RJ, Brasil; e-mail: gestevez@id.uff.br

Michael Yampolsky, Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, Canada; e-mail: yampol@math.toronto.edu

Abstract/Résumé:

We construct a hyperbolic attractor of renormalization of bi-cubic circle maps with bounded combinatorics, with a codimension-two stable foliation.

On construit un attracteur hyperbolique de renormalisation pour une application de cercle bicubique à combinatoire bornée, dont la foliation est de codimension deux.

Keywords: Bi-cubic circle maps, bounded type rotation number, hyperbolicity of renormalization, renormalization operator

AMS Subject Classification: Maps of the circle, Universality; renormalization 37E10, 37E20

PDF(click to download): Hyperbolicity of Renormalization for Bi-cubic Circle Maps with Bounded Combinatorics

Michael Yampolsky,Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4; e-mail: yampol@math.toronto.edu

Abstract/Résumé:

We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.

On développe une théorie de renormalisation pour les homéomorphismes analytiques du cercle à deux points critiques cubiques. On démontre un théorème d’hyperbolicité dans le cadre de renormalisation. Comme base des démonstrations, on développe des bornes complexes a priori pour les applications du cercle dans lui-même aux points critiques multiples

Keywords: Critical circle map, Renormalization, complex bounds

AMS Subject Classification: Maps of the circle, Universality; renormalization, Renormalization 37E10, 37E20, 37F25

PDF(click to download): Renormalization of Bi-cubic Circle Maps

Michael Yampolsky,Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4; e-mail: yampol@math.toronto.edu

Abstract/Résumé:

In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations of such maps.

Dans cet article on généralise la théorie de la renormalisation pour les transformations criticales analytiques du circle à point critical cubique au cas de transformations à exposant critical impair arbitraire, en démontrant une affirmation de rigidité quasi-conforme.

Keywords: Blaschke fractions, Renormalization, critical circle maps, rigidity

AMS Subject Classification: Maps of the circle, Universality; renormalization, Renormalization 37E10, 37E20, 37F25

PDF(click to download): Renormalization of Unicritical Analytic Circle Maps