Metric properties of boundary maps: Hilbert entropy and non-differentiability
Résumé
We interpret the Hilbert entropy of a convex projective structure on a closed surface of higher genus as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space F(R^3). 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 measures.
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