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.

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.

Description

Ramanujan Hall

Date

Mon, April 10, 2017

Start Time

4:00pm-5:00pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

DEFAULT ADMINISTRATOR

Updated

Sun, April 9, 2017 10:04pm IST