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Markov partitions, Martingale and symmetric conjugacy of circle endomorphisms

Author: Yunchun Hu
Journal: Proc. Amer. Math. Soc. 145 (2017), 2557-2566
MSC (2010): Primary 32G15; Secondary 30C99, 30F99, 37F30
Published electronically: December 9, 2016
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Abstract: The main result in this paper is that there is an example of a conjugacy between two expanding Blaschke products on the circle which preserve the Lebesgue measure such that this conjugacy is symmetric at one point but not symmetric on the whole unit circle. Since the proof uses a symmetric rigidity result in a work by Y. Jiang, we use martingale sequences for uniformly quasisymmetric circle endomorphisms developed in an earlier work of the author to give a simple proof. Furthermore, we give a detailed proof of the result in that prior work of the author that the limiting martingale is invariant under symmetric conjugacy.

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Additional Information

Yunchun Hu
Affiliation: Department of Mathematics and Computer Science, Bronx Community College, 2155 University Avenue, Bronx, New York 10453

Keywords: Martingale, quasisymmetric circle homeomorphism, symmetric circle homeomorphism, uniformly quasisymmetric circle endomorphism, uniformly symmetric circle endomorphism, symmetric at one point
Received by editor(s): August 26, 2015
Received by editor(s) in revised form: July 22, 2016
Published electronically: December 9, 2016
Additional Notes: The research was supported by PSC-CUNY Grants.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society