Dynamics Seminar
Paul Apisa
University of Wisconsin Madison
Morse-Smale Dynamics on Surfaces, Geometric Structures, and the Framed Mapping Class Group
Abstract: A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of SL(2, R) on the plane induces an action of SL(2, R) on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. Curiously, an analogue of the Hodge theorem tells us that any vector field on a Riemann surface that vanishes at finitely many points P can be homotoped (through vector fields only vanishing at P) to the straight line foliation of a dilation surface.
The first result that I will present, joint with Nick Salter, produces an SL(2, R)-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a K(pi, 1) where pi is the framed mapping class group.
The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported SL(2, R) invariant measure on the moduli space of dilation surfaces cannot be a finite measure.
Note unusual day
Tuesday November 19, 2024 at 3:00 PM in 636 SEO