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 Poincaré 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.

Version v2 is the electronic copy of the version published in Qualitative Theory of Dynamical Systems

References in corpus (2)

• Dynamical zeta functions for Axiom A flows
• There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus