Renormalization of Bi-cubic Circle Maps

C. R. Math. Rep. Acad. Sci. Canada Vol. 41 (4) 2019, pp. 57-83
(Received: 2019-09-03 , Revised: 2019-12-19)

Michael Yampolsky,Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4; e-mail:


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

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