Teichmuller spaces and HR structures for hyperbolic surface dynamics

Abstract

We construct a Teichmuller space for the C1+-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 C1+-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 C1+-conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmuller space.