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

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 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 measure.