Quasisymmetric distortion and rigidity of expanding endomorphisms of 𝑆¹

de Faria, Edson

doi:10.1090/S0002-9939-96-03218-2

Quasisymmetric distortion and rigidity of expanding endomorphisms of $S^1$
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Proc. Amer. Math. Soc. 124 (1996), 1949-1957 Request permission

Abstract:

In this paper we examine a result of D. Sullivan according to which two $C^{1+\alpha }$ expanding endomorphisms of the circle are $C^{1+\alpha }$ conjugate as soon as they are symmetrically conjugate. We develop general a priori estimates on the local distortion of quasisymmetric mappings and combine them with the classical naive distortion lemma to present a complete proof of Sullivan’s result. A new proof is offered at the end that renders unnecessary the use of Markov partitions or the control of eigenvalues at periodic points.

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