Ratner's property and mild mixing for smooth flows on surfaces

TL;DR

This paper proves that certain special flows over interval exchange transformations with specific singularities exhibit switchable Ratner's property, leading to the conclusion that these flows are mildly mixing but not mixing on higher genus surfaces.

Contribution

It demonstrates that special flows over IETs with symmetric logarithmic singularities satisfy switchable Ratner's property, establishing mild mixing without mixing.

Findings

Flows are mildly mixing but not mixing.

Flows satisfy switchable Ratner's property.

Existence of such flows on higher genus surfaces.

Abstract

Let $T=(T_t^f)_{t\in \mathbb{R}}$ be a special flow built over an IET $T : T \to T$ of bounded type, under a roof function f with symmetric logarithmic singularities at a subset of discontinuities of T. We show that $T$ satisfies so-called switchable Ratner's property. A consequence of this fact is that such flows are mildly mixing. Thus, on each compact, connected, orientable surface of genus greater than one there exist flows which are mildly mixing and not mixing.