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4:00 pm Wednesday, March 11, 2009 Geometry-Analysis Seminar: Finitely-Additive Measures on the Asymptotic Foliations of a Markov Compactumby
Alexander Bufetov (Rice University) in HB 227- The talk will be devoted to the speed of convergence in the ergodic theorem for translation flows on flat surfaces. 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 that grow slower than any power of 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: it is proved that along certain sequences of times ergodic integrals, normalized to have variance one, converge in distribution to a non-degenerate compactly supported measure. 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.
Submitted by damanik@rice.edu |