Metric properties of boundary maps, Hilbert entropy and non-differentiability

2023/10/11 by Beatrice Pozzetti, Pozzetti, Beatrice, Andrés Sambarino +1
Mathematics · Physics and Astronomy · #22E40 #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2310.07373

openalex publication_date 2023/10/11 · openalex created_date 2023/10/13 · openalex updated_date 2026/07/28

Abstract

We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space \mathcal F(\mathbb R³). Generalizations for regularity properties of boundary maps between locally conformal representations are also discussed. An ingredient for the proofs is the concept of hyperplane conicality that we introduce for a θ-Anosov representation into a reductive real-algebraic Lie group G. In contrast with directional conicality, hyperplane-conical points always have full mass for the corresponding Patterson-Sullivan measure.

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