Closed Geodesics in Dilation Surfaces
Time: 2025-01-03
Published By: Xiaoni Tan
Speaker(s): Guillaume Tahar (BIMSA)
Time: 13:00-15:00 January 4, 2025
Venue: Lecture Hall, Jiayibing Building, Jingchunyuan 82, BICMR
Abstract:
Dilation surfaces are generalizations of translation surfaces in which the transition maps of the atlas are both translations and homotheties. Just as translation surfaces arise as suspensions of Interval Exchange Maps, dilation surfaces arise as suspensions of Affine Interval Exchange Maps.
In this talk, we will prove that all dilation surfaces contain periodic trajectories. The proof relies on an analysis of the possible degenerations of dilation structures at the boundary of the moduli space, which is made more challenging by the absence of an invariant notion of length in this geometric setting.
We will provide the necessary background on translation and dilation surfaces, the associated dynamical systems, as well as the structure of their moduli spaces.
