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

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.