Superdensity and bounded geodesics in moduli space

Abstract

Following Beck-Chen, we say a flow $ϕ_t$ 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 $\{ϕ_t x\}_{0 \le t \le cT}$ is $1/T$-dense in $X$. We show that a linear flow on a translation surface is superdense if the associated Teichmüller geodesic is bounded. Conversely, if the linear flow is superdense, we show that along the Teichmüller 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.