Low dimensional geometric dynamics

Minimal rays on closed surfaces

speaker: Jan Philipp Schröder (Ruhr-Universität Bochum)

abstract: We consider any Finsler metric on a closed, orientable surface of genus at least one. Morse (1924) and Hedlund (1932) proved that minimal geodesic rays in the universal cover move asymptotically towards a single point at infinity (in the Poincaré disc: a point on the unit circle). Moreover, they described the finer structure of the set of rays whose asymptotic direction is the endpoint of an axis in the universal cover. Bangert (1988) proved in the case of genus one, i.e. for the torus, that rays with a common asymptotic direction cannot intersect transversely, provided the asymptotic direction is not the endpoint of an axis (i.e. has irrational slope). In this talk, I will generalize Bangert's result to the higher genus case, thus completing the study of intersections of rays in closed surfaces. As an application of our result, we characterize the boundary at infinity of the universal cover defined by Gromov using the Finsler distance function.

timetable:
Mon 20 Apr, 11:30 - 12:30, Aula Dini
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