Asymptotic rigidity of scaling ratios for critical circle mappings

Abstract

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 quasiconformal surgical 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 a 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 $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ 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.