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4:00 pm Wednesday, April 14, 2010 Geometry-Analysis Seminar: Limit Theorems for Translation Flowsby
Alexander Bufetov (Rice University) in HB 227- 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
Submitted by damanik@rice.edu |