Description
Title:
Skew Products Over The Irrational Rotation, The Central Limit Theorem And RATs
Abstract:
Let f be a step function on the circle with zero mean and rational discontinuities while alpha is a quadratic irrational. The point-wise ergodic theorem tells us that the ergodic sums, f(x)+f(x+alpha)+...+f(x+(n-1)alpha) is o(n) for almost every x but says nothing about its deviations from zero, that is, its discrepancy; the study of these deviations naturally draws us to the study of ergodic transformations on infinite measure spaces, viz., skew products over irrational rotations. In this talk, after a brief introduction to these terms, we will learn how the temporal statistics of the ergodic sums for x=0 can be studied via random affine transformations (RATs) leading to a central limit theorem and other fine properties like the visit times to a neighbourhood of 0 vis-à-vis bounded rational ergodicity (all of course time permitting). This is reporting on joint work with Jon Aaronson and Michael Bromberg.