The research is a study of dynamical properties of translation flows on flat surfaces, with a special emphasis on connections with probability theory. In previously funded work, the proposer has obtained new limit theorems for translation flows. These results give a completely new type of limit theorems in ergodic theory and open a wide array of questions. The proposer will continue his study of limit distributions for translation flows. An important particular case is that of flows along stable foliations of pseudo-Anosov diffeomorphisms.A specific tool essential for this study is the symbolic coding, developed and studied by the proposer, of translation flows as suspension flows over Vershik automorphisms, a construction developing earlier work of S.Ito. Strongly chaotic, hyperbolic behavior of the Teichmueller flow controls the mildly chaotic, parabolic behavior of translation flows.In the second, more geometric, part of the project, the proposer will continue his investigation of the chaotic properties of the Teichmueller geodesic flow.
We see chaotic behavior in the evolution of stock prices, weather patterns, turbulent fluids and traffic jams. How to control chaotic behavior? A key role in our modern perception of chaos is played by the ergodic theory of dynamical systems. The proposed research studies the slow chaos for flows on surfaces, a central class of examples in geometry and physics. The project also has an important pedagogical component. Through lectures by visiting scholars, working seminars and reading projects related to the proposal, the proposer will continue involving Rice undergraduate and graduate students in Mathematical research.
