Logarithmic bounds for ergodic sums of certain flows on the torus: a short proof

TL;DR

This paper provides a concise proof that ergodic sums for certain smooth flows on the torus grow at most logarithmically from linear, using circle rotation properties and the Denjoy-Koksma inequality.

Contribution

It introduces a short, elegant proof for logarithmic bounds on ergodic sums of specific torus flows with constant type rotation numbers.

Findings

Ergodic sums grow at most logarithmically from linear for the considered flows.

The proof relates torus flow sums to circle Birkhoff sums and applies Denjoy-Koksma inequality.

An example of a nonminimal flow satisfying the assumptions is provided.

Abstract

We give a short proof that the ergodic sums of $\mathcal{C}^1$ observables for a $\mathcal{C}^1$ flow on $\mathbb{T}^2$ admitting a closed transversal curve whose Poincar\'e map has constant type rotation number have growth deviating at most logarithmically from a linear one. For this, we relate the latter integral to the Birkhoff sum of a well-chosen observable on the circle and use the Denjoy-Koksma inequality. We also give an example of a nonminimal flow satisfying the above assumptions.