Edson de Faria

Rigidity of critical circle mappings I

Abstract.We prove that two C3 critical circle maps with the same rotation number in a special set ? are C1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C0 sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C∞ critical circle maps with the same rotation number that are not C1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.

Rigidity of critical circle mappings II

We prove that two C3 critical circle maps with the same rotation number in a special set ? are C1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C0 sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C∞ critical circle maps with the same rotation number that are not C1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers.

ASYMPTOTIC RIGIDITY OF SCALING RATIOS FOR CRITICAL CIRCLE MAPPINGS

Let

Limit laws of entrance times for homeomorphisms of the circle

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Quasisymmetric distortion and rigidity of expanding endomorphisms of

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

On Sloane’s Persistence Problem

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Real bounds and Lyapunov exponents

about the critical point are asymptotically independent of

ON CONFORMAL DISTORTION AND SULLIVAN'S SECTOR THEOREM

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Beau bounds for multicritical circle maps

. 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

Differentiability of correlations in realistic quantum mechanics

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