For more information about this meeting, contact Anatole Katok.

Title: | Livshits - type theorems and applications, I. |

Seminar: | Working Seminar: Dynamics and its Working Tools |

Speaker: | Boris Kalinin, University of South Alabama, visiting Penn State |

Abstract: |

A. Livshits pioneered the study of cohomology for hyperbolic
dynamical systems in the early 1970’s. The simplest question in
this area is that of existence and regularity of a solution g(x) to
a cohomological equation f (x) = g(T x)− g(x), where T : M → M is
a hyperbolic dynamical system, say a transitive Anosov
diﬀeomorphism, and f is a given function on M , usually Holder
continuous or smooth. Equations of this type appear naturally
in many areas of dynamics. For example, if f (x) is the logarithm
of the Jacobian of T then solving this equation is equivalent to
ﬁnding an absolutely continuous invariant measure for T .
We will recall the fundamental results obtained by Livshits: ex-
istence of a continuous solution when the natural periodic ob-
structions vanish and continuity of any measurable solution. We
will then turn to similar questions in a more general and diﬃcult
setting when the functions f and g take values in a group G. |

### Room Reservation Information

Room Number: | MB216 |

Date: | 09 / 08 / 2009 |

Time: | 03:30pm - 06:00pm |