Anosov representations and dominated splittings

TL;DR

This paper establishes a connection between Anosov representations and dominated splittings, providing new characterizations, recovering recent results, and offering a novel proof of the higher rank Morse Lemma.

Contribution

It introduces a new link between Anosov representations and dominated splittings, leading to alternative characterizations and proofs of key results.

Findings

Equivalent characterizations for Anosov representations.

Recovery of recent results by Guéritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti.

A new proof of the higher rank Morse Lemma.

Abstract

We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Gu\'eritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti.