Harry Bray

Mathematical Sciences Department
George Mason University
hbray [at] gmu [dot] edu

Research

My research interests are in geometry, topology, and dynamical systems. More specifically, I am interested in geodesic flows, geometric structures on manifolds, Hilbert geometries, convex projective structures, Patterson-Sullivan theory, ergodic theory, thermodynamic formalism, symbolic dynamics, (nonuniformly) hyperbolic dynamical systems, and (rel) hyperbolic spaces and groups.

I have also studied the Thurston set and Thurston Master Teapot. Check out my collaborator Giulio Tiozzo's visualizations of the Master Teapot. Here is a Quanta article on the Thurston set and related topics.

Here are pictures of objects I've studied, made by my students: [Thurston's master teapot] [the Thurston set] [deformations of convex projective triangle groups] [Benoist 3-manifold]

Preprints and Papers

• Pressure metrics for cusped Hitchin components.
  Joint with Dick Canary, Nyima Kao, and Giuseppe Martone.
  [arXiv]
• A global shadow lemma and logarithm law for geometrically finite Hilbert geometries.
  Joint with Giulio Tiozzo
  [arXiv]
  [lecture notes on Khinchin theorem and logarithm law]
• Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations.
  Joint with Dick Canary, Nyima Kao, and Giuseppe Martone.
  Journal für die reine und angewandte Mathematik (Crelles Journal), vol. 2022, no. 791, 2022, pp. 1-51.
  [journal][arXiv]
• Entropy rigidity for finite volume strictly convex projective manifolds.
  Joint with Dave Constantine
  Geometrie Dedicata (2021) Volume 214, 543-557
  [journal] [arXiv]
• Pressure metrics for deformation spaces of quasifuchsian groups with parabolics.
  Joint with Dick Canary and Nyima Kao
  accepted at Algebraic & Geometric Topology
  [arXiv]
• The Shape of Thurston's Master Teapot.
  Joint with Diana Davis, Kathryn Lindsey, and Chenxi Wu
  Advances in Mathematics (2021) Volume 377, 107481
  [journal] [arXiv]
• Ergodicity of Bowen-Margulis measure for the Benoist 3-manifolds.
  Journal of Modern Dynamics (2020) Volume 16, 305–329
  [journal][arXiv]
• Geodesic flow of nonstrictly convex Hilbert geometries.
  Annales de l’Institut Fourier (2020) Volume 70: 4 1563-1593
  [journal][arXiv]
• Entropy rigidity and Hilbert volume.
  Joint with Ilesanmi Adeboye and Dave Constantine
  Discrete Contin. Dyn. Syst. Ser. A (2019) no. 39(4): 1731-1744. doi: 10.3934\/dcds.2019075.
  [journal] [arXiv]