arXiv subjects
Publications and source records attributed to Klaus Dankwart.
Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.
We investigate typical behavior of geodesics on a closed flat surface $S$ of genus $g\geq 2$. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same conformal class. This quotient is asymptotically constant $F$ a.e. We show that $F$ is bounded from below by the inverse of the volume entropy $e(S)$. Moreover, we construct a geodesic flow together with a measure on $S$ which is induced by the Hausdorff measure of the Gromov boundary of the universal cover. Denote by $e(S)$ the volume entropy of $S$ and let $c$ be a compact geodesic arc which connects singularities. We show that a typical geodesic passes through $c$ with frequency that is comparable to $\exp(-e(S)l(c))$. Thus a typical bi-infinite geodesic contains infinitely many singularities, and each geodesic between singularities $c$ appears infinitely often with a frequency proportional to $\exp(-e(S)l(c))$.
We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infinity. Moreover, we estimate the entropy of a locally isometric branched covering of a flat surface by the entropy of base surface and the geometry of the covering map.