Correlation of the renormalized Hilbert length for convex projective surfaces

Abstract

In this paper we focus on dynamical properties of (real) convex projective surfaces. Our main theorem provides an asymptotic formula for the number of free homotopy classes with roughly the same renormalized Hilbert length for two distinct convex real projective structures. The correlation number in this asymptotic formula is characterized in terms of their Manhattan curve. We show that the correlation number is not uniformly bounded away from zero on the space of pairs of hyperbolic surfaces, answering a question of Schwartz and Sharp. In contrast, we provide examples of diverging sequences, defined via cubic rays, along which the correlation number stays larger than a uniform strictly positive constant. In the last section, we extend the correlation theorem to Hitchin representations.

arXiv comment27 pages. improved exposition. Comments welcome

References in corpus (9)

• Hyperconvex representations and exponential growth
• Cyclic Higgs bundles and the affine Toda equations
• Survey on Affine Spheres
• The degeneration of convex RP^2 structures on surfaces
• On the Hilbert geometry of simplicial Tits sets
• Currents, Systoles, and Compactifications of Character Varieties
• Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$
• Infinitesimal Zariski closures of positive representations
• The embedding of the space of negatively curved surfaces in geodesic currents

Cited by in corpus (1)

• Correlation Number for Potentials with Entropy Gaps and Cusped Hitchin Representations