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

TL;DR

This paper links the Hilbert entropy of convex projective structures on higher-genus surfaces to the Hausdorff dimension of non-differentiability points in the limit set, introducing hyperplane conicality for Anosov representations.

Contribution

It introduces the concept of hyperplane conicality for Anosov representations and relates Hilbert entropy to the Hausdorff dimension of non-differentiability points.

Findings

Hilbert entropy equals the Hausdorff dimension of non-differentiability points.

Hyperplane conicality points have full Patterson-Sullivan measure.

Generalizations for boundary map regularity are discussed.

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 $\mathcal F(\mathbb R^3)$. 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 $\theta$-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.