Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms

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July 1999, 5(3): 535-552. doi: 10.3934/dcds.1999.5.535

Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms

Guizhen Cui ^1, , Yunping Jiang ^2, and Anthony Quas ^3,

1.	Institute of Mathematics, Academia Sinica, Beijing 100080, China
2.	Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, United States
3.	Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

Received October 1997 Revised March 1999 Published May 1999

Abstract
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We study the scaling function of a $C^{1+h}$ expanding circle endomorphism. We find necessary and sufficient conditions for a Hölder continuous function on the dual symbolic space to be realized as the scaling function of a $C^{1+h}$ expanding circle endomorphism. We further represent the Teichmüller space of $C^{1+h}$ expanding circle endomorphisms by the space of Hölder continuous functions on the dual symbolic space satisfying our necessary and sufficient conditions and study the completion of this Teichmüller space in the universal Teichmüller space.

Keywords: Scaling function, Teichmüller space., g-measure.

Mathematics Subject Classification: 58F23, 30C6.

Citation: Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535

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