{"paper":{"title":"Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\\mathrm{PSL}(3,\\mathbb{R})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.DS","math.GT"],"primary_cat":"math.MG","authors_text":"Nicolas Tholozan","submitted_at":"2015-06-15T15:50:14Z","abstract_excerpt":"We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, \\[d^B(x,y) < d^H(x,y) +1.\\] We obtain two interesting consequences: the first one is the volume entropy rigidity for Hilbert geometries : for any proper convex domain of $\\mathbb{R}\\mathbf{P}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation of a surface group into $\\mathrm{PSL}(3,\\mathbb{R})$, there exists a Fuchsian representation $j$ in $\\mathrm{PSL}(2,\\mathbb{R})$ s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04640","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}