Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms

Pages: 535 - 552, Volume 5, Issue 3, July 1999

doi:10.3934/dcds.1999.5.535       Abstract        Full Text (258.6K)

Guizhen Cui - Institute of Mathematics, Academia Sinica, Beijing 100080, China (email)
Yunping Jiang - Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, United States (email)
Anthony Quas - Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States (email)

Abstract: 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, g-measure, Teichmüller space.
Mathematics Subject Classification:  58F23, 30C62.

Received: October 1997;      Revised: March 1999;      Published: May 1999.