Logarithmic bounds for ergodic sums of certain flows on the torus: a short proof (Q2156332)

Consider a \(\mathcal{C}^1\) flow \(h_t\) defined on the 2-dimensional torus \(\mathbb{T}^2\). The author studies the growth of the \textit{ergodic integral} of a \(\mathcal{C}^1\) observable \(f:\mathbb{T}^2\longrightarrow\mathbb{R}\) at \(x\in\mathbb{T}^2\) and \(T>0\). This is the quantity \[ H_{x,T}(f)=\int_0^T f\circ h_t(x)dt. \] It is well known that if \(h_t\) has neither critical points nor periodic trajectories, then \(h_t\) is smoothly conjugated to the suspension of the first return map of a simple closed curve transverse to \(h_t\). Moreover, \(h_t\) is uniquely ergodic. A short proof of this fact is given for the sake of completeness (Theorem 2.1). The main result of the paper is Theorem 2.2 which states, roughly speaking, that, in the previous situation, the ergodic integral of any observable with zero average with respect to the unique invariant measure of \(h_t\) grows at most logarithmically, provided the rotation number of the first return map has finite continuous fraction expansion. The last section of the paper is devoted to proving Theorem 3.1 that states that not every flow in the conditions of Theorem 2.2 is minimal.