Patterson–Sullivan theory and equidistribution in Hilbert geometry

Group Actions, Geometry and Dynamics
Event time: 
Thursday, October 5, 2023 - 4:00pm
Pierre-Louis Blayac
Speaker affiliation: 
University of Michigan
Event description: 
 In the theory of discrete subgroups of Lie groups, given a length function on the Lie group G, one popular object of study is the asymptotic when R goes to infinity of the number of loxodromic elements with length less R in a discrete subgroup H, and the distribution of the fixed points of these loxodromic elements.
When G is the group of (real) projective transformations and H acts properly discontinuously and cocompactly on a strictly convex domain O of the projective space, Yves Benoist noticed that one can use the Hilbert metric of O, and the associated Hilbert geodesic flow, to estimate the above counting function, for a special length function called the Hilbert length.
An important ingredient in Benoist’s result is the uniform hyperbolicity of the Hilbert geodesic flow, which does not hold when O is not strictly convex.
In this talk we will develop and use instead the theory of Patterson–Sullivan measures for Hilbert geometries O/H that satisfy a mild rank-one assumption, and are not necessarily strictly convex nor compact.
This will yield counting results for H in the cases where the induced Bowen–Margulis measure is finite. (This is joint work with Feng Zhu).