Closed geodesics in dilation surfaces

Abstract

We introduce the notion of exotic dilation surface, which provides a way to produce a weak form of compactification of the moduli space of dilation surfaces. This notion is devised to enable one to keep track of dynamical properties of their directional foliations along degenerating sequences. We derive from this construction a proof that every dilation surface contains a closed regular geodesic.

Horizon saddle connections and Morse–Smale dynamics of dilation surfaces

“…A weak version of Conjecture 1.2 has been proved in [1]. It asserts that directions of closed geodesics are dense in S 1 .…”

“…A weak version of Conjecture 1.2 has been proved in [1]. It asserts that directions of closed geodesics are dense in S 1 . Unfortunately, no information on whether a significant part of these geodesics are hyperbolic is provided.…”