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Ergodicity and equidistribution in Hilbert geometry
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abstract
We show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT($-1$) or rank-one CAT($0$) spaces, also hold for rank-one properly convex projective structures, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective structures. More specifically, with respect to the Sullivan measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.
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On 4-dimensional convex projective domains invariant by a lattice of $\mathrm{SL}_2 (\mathbb{R})$
Four counterexamples to converses of geometric-finiteness implications in round convex projective geometry are constructed using 4-dimensional domains invariant under the irreducible representation of SL₂(ℝ).
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