Correlation of the renormalized Hilbert length for convex projective surfaces

TL;DR

This paper investigates the asymptotic behavior of the correlation of renormalized Hilbert lengths on convex projective surfaces, revealing new insights into their dynamical properties and extending results to Hitchin representations.

Contribution

It provides an asymptotic formula for correlation numbers of convex projective structures and extends the correlation theorem to Hitchin representations.

Findings

Correlation number is not uniformly bounded away from zero for pairs of hyperbolic surfaces.

Diverging sequences can have correlation numbers bounded away from zero.

The correlation theorem is extended to Hitchin representations.

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.