Uniformly hyperbolic finite-valued $\mathrm{SL}(2,ℝ)$-cocycles

Artur Avila; Jairo Bochi; Jean-Christophe Yoccoz

doi:10.4171/cmh/212

JournalscmhVol. 85, No. 4pp. 813–884

Uniformly hyperbolic finite-valued $SL (2, R)$ -cocycles

Artur Avila
Université Pierre et Marie Curie, Paris, France
Jairo Bochi
PUC-Rio, Rio de Janeiro, Brazil
Jean-Christophe Yoccoz
Collège de France, Paris, France

$Uniformly hyperbolic finite-valued $\mathrm{SL}(2,ℝ)$-cocycles cover$

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Abstract

We consider finite families of $SL (2, R)$ matrices whose products display uniform exponential growth. These form open subsets of $SL (2, R)^{N}$ , and we study their components, boundary, and complement. We also consider the more general situation where the allowed products of matrices satisfy a Markovian rule.

Cite this article

Artur Avila, Jairo Bochi, Jean-Christophe Yoccoz, Uniformly hyperbolic finite-valued $SL (2, R)$ -cocycles. Comment. Math. Helv. 85 (2010), no. 4, pp. 813–884

DOI 10.4171/CMH/212