Temporal distributional limit theorem for cocycles over rotations
Ergodic Theory and Dynamical Systems Seminar
16th November 2017, 3:00 pm – 4:00 pm
Howard House, 4th Floor Seminar Room
For a measure preserving system (X,B,μ,T ) and a real valued function f on X, temporal random variables along an orbit of a fixed point x in X are obtained by considering the Birkhoff sums Sn(f,x), n = 1,...,N and choosing n randomly uniformly from 1, ..., N. These r.v’s, measure the fraction of time that Birkhoff sums spend in various sets. If, under proper normalization, as N tends to infinity, these variables converge to a non-trivial distribution, we say that f satisfies a temporal limit theorem along the orbit of x (when the limit is Gaussian, we refer to this as temporal CLT). The aim of the talk is to introduce the relevant concepts and sketch a proof of a temporal CLT for piecewise constant cocycles with a single breakpoint, over an irrational rotation with a badly approximable rotation number. This result generalises earlier results by J.Beck and by D.Dolgopyat and O.Sarig. This is joint work with C.Ulcigrai.