Markov partitions, Martingale and symmetric conjugacy of circle endomorphisms

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.