Quasisymmetric distortion and rigidity of expanding endomorphisms of

In this paper we examine a result of D. Sullivan according to which two expanding endomorphisms of the circle are 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.