Superdensity and bounded geodesics in moduli space

TL;DR

This paper investigates the conditions under which linear flows on translation surfaces are superdense, linking superdensity to boundedness of Teichmüller geodesics, and generalizes previous results on lattice surfaces.

Contribution

It establishes a bidirectional relationship between superdensity of linear flows and bounded Teichmüller geodesics on translation surfaces, extending prior work on lattice surfaces.

Findings

Superdensity implies bounded Teichmüller geodesics.

Bounded geodesics imply superdensity of flows.

Generalizes Beck-Chen's results to broader translation surfaces.

Abstract

Following Beck-Chen, we say a flow $\phi_t$ on a metric space $(X, d)$ is superdense if there is a $c > 0$ such that for every $x \in X$, and every $T>0$, the trajectory $\{\phi_t x\}_{0 \le t \le cT}$ is $1/T$-dense in $X$. We show that a linear flow on a translation surface is superdense if the associated Teichm\"uller geodesic is bounded. Conversely, if the linear flow is superdense, we show that along the Teichm\"uller geodesic, the diameter of the surface remains bounded. This generalizes work of Beck-Chen on lattice surfaces, and is reminiscent of work of Masur on unique ergodicity.