Thurston's cataclysms for Anosov representations

TL;DR

This paper generalizes Thurston's cataclysm deformations from hyperbolic structures to Anosov representations, constructing shear deformations along geodesic laminations with flag decorations and analyzing their geometric properties.

Contribution

It introduces a new class of shear deformations for Anosov representations along geodesic laminations, extending Thurston's classical cataclysm concept.

Findings

Constructed shear deformations for Anosov representations

Established a variation formula for length functions

Proved geometric properties of the deformations

Abstract

Given an Anosov representation $\rho \colon \pi_1(S) \to \PSL_{n}(\mathbb{R})$ and a maximal geodesic lamination $\lambda$ in a surface $S$, we construct shear deformations along the leaves of the geodesic lamination $\lambda$ endowed with a certain flag decoration, that is provided by the associated flag curve $\mathcal{F}_\rho\colon \Sinf \to \mathrm{Flag}(\mathbb{R}^n)$ of the Anosov representation $\rho$; these deformations generalize to Labourie's Anosov representations Thurston's cataclysms for hyperbolic structures on surfaces. A cataclysm is parametrized by a transverse $n$--twisted cocycle for the orientation cover $\La$ of $\lambda$. In addition, we establish various geometric properties for these deformations. Among others, we prove a variation formula for the associated length functions $\ell^i_\rho$ of the Anosov representation $\rho$.