Superdensity and bounded geodesics in moduli space

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.