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We interpret the Hilbert entropy of a convex projective structure on a closedhigher-genus surface as the Hausdorff dimension of the non-differentiabilitypoints of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.Generalizations for regularity properties of boundary maps between locallyconformal representations are also discussed. An ingredient for the proofs isthe concept of hyperplane conicality that we introduce for a $\theta$-Anosovrepresentation into a reductive real-algebraic Lie group $G$. In contrast withdirectional conicality, hyperplane-conical points always have full mass for thecorresponding Patterson-Sullivan measure.

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