# American Institute of Mathematical Sciences

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

## Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms

 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

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