Correlation of the renormalized Hilbert length for convex projective surfaces

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 i...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 43; no. 9; pp. 2938 - 2973
Main Authors DAI, XIAN, MARTONE, GIUSEPPE
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2023
Subjects
Asymptotic properties
Correlation
Entropy
Original Article
Sequences
Theorems
low-dimensional dynamics
higher Teichmüller theory
37D35
57M50
thermodynamic formalism
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2022.56