The subject of this lecture is the ongoing research
with Hugo Parlier about the distribution of the complete simple geodesics on
a compact Riemann surface endowed with the Poincare metric of constant
cuvrvature -1. It is well known that the complete geodesics on such a
surface are dense. The situation is quite different, however, if one restricts one's
consideration to the simple ones, that is, those without self intersections. Birman and Series, in 1985, showed that the collection formed by the latter is nowhere dense. Our goal is to give a quantitative version of this result and to visualize the domains on the surface into which the complete simple geodesics never enter.


The lecture is accompanied by numerous pictures.