Circle averages and disjointness in typical flat surfaces on every Teichmueller disc

TL;DR

This paper demonstrates that on typical translation surfaces, flows in almost every pair of directions are disjoint and not isomorphic, and it applies these results to the convergence of circle averages, revealing new properties of such surfaces.

Contribution

It establishes disjointness of flows in almost every pair of directions on typical translation surfaces and applies this to circle average convergence, a previously unresolved issue.

Findings

Flows in almost every pair of directions are disjoint.

Disjointness property holds beyond torus covers.

Convergence of circle averages is proven for these surfaces.

Abstract

We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with this property. We provide an application to the convergence of `circle averages' for the flow (away from a sequence of radii of density 0) for such surfaces. Even the density of a sequence of 'circles' was only known in a few special examples.