Disjointness of rescalings of smooth area preserving flows on surfaces

Abstract

We consider the problem of \emph{disjointness of rescalings} $(φ_{κt})_{t\in \mathbb{R}}$, $κ\in\mathbb{R}$ of a flow $(φ_{t})_{t\in \mathbb{R}}$ in the context of smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces. We show that, when the genus of the surface is $g\ge 2$, almost every locally Hamiltonian flow with 2g-2 non-degenerate simple saddles is such that any distinct two rational rescalings $(φ_{κt})_{t\in \mathbb{R}}$ and $(φ_{κ' t})_{t\in \mathbb{R}}$ with $κ=p/q$ and $κ'=p'/q'$ of different absolute values, are disjoint. Previous results on disjointness of rescalings were available only for rescalings for locally Hamiltonian flows and their special flow representations in genus one. The result is proved using a criterion for disjointness based on the study of the distribution of Birkhoff sums of a special representation and in particular estimates on their exponential tails decay. A key novel geometric ingredient in the proof is the existence of a sequence of rigidity times which display what we call bounded-type rigidity, so that a large set of points comes back in time $q$ with distance $O(1/q)$. To produce such bounded-type rigidity times we exploit a particular way of degeneration of a translation surface to a flat torus for which the vertical flow has bounded-type rotation number.