Abstract

The general theory of locally homogeneous geometric structures (flat Cartan connections) originated with Ehresmann. Their classification is analogous to the classification of Riemann surfaces by the Riemann moduli space. In general, however, the analog of the moduli space is intractable, but leads to a rich class of dynamical systems.  

For example, classifying Euclidean geometries on the torus leads to the usual action of the SL(2,Z)  on the upper half-plane. This action is dynamically trivial, with a quotient space the familiar modular curve.  In contrast, the classification of other simple geometries on  on the torus leads to the standard linear action of SL(2,Z) on R<sup>2,  with chaotic dynamics and a pathological quotient space.  

This talk describes such dynamical systems, and we combine Teichmueller theory to understand the geometry of the moduli space when the topology is enhanced with a  conformal structure. In joint work with Forni, we prove the corresponding extended Teichmueller flow is strongly mixing.  

Basic examples arise when  the moduli space  is described by the nonlinear symmetries of cubic equations like Markoff’s equation x</sup>2 y² z2 = x y z.  Here both trivial and chaotic dynamics arise simultaneously, relating to possibly singular hyperbolic-geometry structures on surfaces. (This represents joint work with McShane-Stantchev- Tan.)