Ergodic Theory and Dynamical Systems

Teichmüller spaces and HR structures for hyperbolic surface dynamics

A. A. PINTO a1 and D. A. RAND a2
a1 DMA, Faculdade de Ciências, Universidade do Porto, 4000 Porto, Portugal (e-mail:
a2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (e-mail:


We construct a Teichmüller space for the C^{1+}-conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structure with each of the stable and unstable laminations, we show that there is a one-to-one correspondence between these HR structures and the C^{1+}-conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C^{1+}-conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmüller space.

(Received June 23 2000)
(Revised January 9 2002)