For more information about this meeting, contact Anatole Katok, Hope Shaffer, Yakov Pesin, Dmitri Burago.

Title: | Limit Theorems for Translation Flows |

Seminar: | Center for Dynamics and Geometry Seminars |

Speaker: | Alexander Bufetov, Rice University and Steklov Institute |

Abstract: |

Consider a compact oriented surface of genus at least two
endowed with a holomorphic one-form.
The real and the imaginary parts of the one-form define two foliations on the
surface, and each
foliation defines an area-preserving translation flow. By a Theorem of H.Masur
and W.Veech,
for a generic surface these flows are ergodic. The talk will be devoted to
the speed of convergence in the ergodic theorem for translation flows.
The main result, which extends earlier work of A.Zorich and G.Forni,
is a multiplicative asymptotic expansion for time averages of Lipschitz
functions.
The argument, close in spirit to that of G.Forni, proceeds by
approximation of ergodic integrals by special holonomy-invariant
Hoelder cocycles on trajectories of the flows.
Generically, the dimension of the space of holonomy-invariant
Hoelder cocycles is equal to the genus of the surface, and the ergodic integral
of a Lipschitz function can be approximated by such a cocycle up to terms
growing slower than any power of the time.
The renormalization effectuated by the Teichmueller geodesic flow
on the space of holonomy-invariant Hoelder cocycles allows one also to obtain
limit theorems
for translation flows.
The argument uses a symbolic representation of translation flows as suspension
flows over
Vershik's automorphisms, a construction similar to one proposed by S.Ito.
The talk is based on the preprint http://arxiv.org/abs/0804.3970 |

### Room Reservation Information

Room Number: | MB106 |

Date: | 03 / 22 / 2010 |

Time: | 03:30pm - 05:30pm |