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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℱ</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> -Anosov representation into a reductive real-algebraic Lie group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="sans-serif">G</mml:mi> </mml:math> . In contrast with directional conicality, hyperplane-conical points always have full mass for the corresponding Patterson–Sullivan measure.