Strictly convex Hilbert geometries naturally generalize constant
negatively curved Riemannian geometries, and the geodesic flow on
quotients has been well-studied by Benoist, Crampon, Marquis, and
others. In contrast, nonstrictly convex Hilbert geometries in three
dimensions have the feel of nonpositive curvature, but also have a
fascinating geometric irregularity which forces the geodesic flow to
avoid direct application of existing nonuniformly hyperbolic theory.
In this talk, we present a geometric approach to studying the geodesic
flow in this setting, culminating in a measure of maximal entropy
which is ergodic for the geodesic flow.