Tremors and horocycle dynamics on the moduli space of transition surfaces

Abstract

We introduce a `tremor’ deformation on strata of translation surfaces. Using it, we give new examples of behaviors of horocycle flow orbits $U_q$ in strata of translation surfaces. In the genus 2 stratum $\mathcal{H}(1,1)$ we find orbits $U_q$ which are generic for a measure whose support is strictly contained in $\overline{U_q}$ and find orbits which are not generic for any measure. We also describe a horocycle orbit-closure whose Hausdorff dimension is not an integer.