# Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

## TL;DR

This paper establishes explicit uniform convergence rates for Birkhoff averages in certain ergodic toral maps, linking the rate to the observable's continuity and the transformation's arithmetic properties, with near-optimal results in one dimension.

## Contribution

It provides explicit convergence rate estimates for Birkhoff averages in specific ergodic toral maps, highlighting the dependence on continuity and arithmetic properties.

## Key findings

- Convergence rates depend on the modulus of continuity and arithmetic properties.
- Estimates are nearly optimal for one-dimensional torus translations.
- Results apply to affine skew product toral transformations.

## Abstract

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one dimensional torus translation, these estimates are nearly optimal.