Law of large numbers for certain cylinder flows

TL;DR

This paper constructs new ergodic cylinder flows based on irrational rotations, demonstrating a law of large numbers with Gaussian distribution for visits to zero, advancing understanding of statistical properties in dynamical systems.

Contribution

It introduces explicit examples of ergodic cylinder flows that satisfy a law of large numbers, with detailed calculations of visit distributions to zero.

Findings

Cylinder flows are ergodic and rationally ergodic along subsequences.

Number of visits to zero follows a Gaussian distribution.

Demonstrates law of large numbers in new dynamical system examples.

Abstract

We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a gaussian distribution.