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 quasi-conformal surgery 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 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 → z +θ − 1 2π sin 2πz, θ 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.Mathematics Subject Classification (1991): Primary 58F03, 30F60, 58F23, 32G05.

Rigidity of critical circle mappings II

“…In other words, the first renormalization of the critical commuting pair of Γ extends in a natural way to a holomorphic pair R(Γ) with the same co-domain V. For the careful construction of R(Γ), see Prop. 2.3 in [2]. There is also a pull-back theorem for holomorphic pairs.…”

“…The concept of holomorphic commuting pair was introduced in [2], and will play a crucial role in this paper. We recall the definition and some of the relevant facts about these objects; henceforth called simply holomorphic pairs.…”

“…Examples of holomorphic pairs with arbitrary rotation number and arbitrary height were carefully constructed in §4 of [2], with the help of the family of entire maps given by…”

“…In this case, consider ζ = f −qm (ζ) and ζ = f −qm+2 (ζ ) and the corresponding interval J = f −qm−qm+2 (J −i ), and apply Lemma 3.7. Then either ζ ∈ D m+1 , in which case the induction step is complete, or else dist (ζ , I m+1 ) ≤ C|I m | and angle (ζ , J ) ≥ ε 1 , in which case we can apply the same argument leading to (2) and (3).…”

“…We use the real a-priori bounds ofŚwiatek and Herman (see [14], [6] and [3]) in the form stated in §2. The complex-analytic part we develop here combines some of the powerful new ideas on conformal rigidity and renormalization recently developed by McMullen in [9], [10], [11] with the basic theory of holomorphic commuting pairs introduced in [2] and a generalization of the Lyubich-Yampolsky approach to the complex bounds given in [8], [15].…”