# Rational ergodicity of Step function Skew Products

## TL;DR

This paper investigates the ergodic properties of step function skew products over circle rotations, demonstrating rational ergodicity for quadratic irrational rotation numbers through analysis of associated affine random walks.

## Contribution

It establishes rational ergodicity of step function skew products over certain rotations, especially for quadratic irrationals, linking dynamics to affine random walk behavior.

## Key findings

- Proves ergodicity for these skew products.
- Shows bounded rational ergodicity for quadratic irrationals.
- Connects orbit statistics to affine random walk models.

## Abstract

We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk.