A uniform dichotomy for generic SL ( 2 , ℝ ) cocycles over a minimal base

Artur Avila; Jairo Bochi

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 3, page 407-417
  • ISSN: 0037-9484

Abstract

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We consider continuous SL ( 2 , ℝ ) -cocycles over a minimal homeomorphism of a compact set K of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

How to cite

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Avila, Artur, and Bochi, Jairo. "A uniform dichotomy for generic ${\rm SL}(2,{\mathbb {R}})$ cocycles over a minimal base." Bulletin de la Société Mathématique de France 135.3 (2007): 407-417. <http://eudml.org/doc/272427>.

@article{Avila2007,
abstract = {We consider continuous $\{\rm SL\}(2,\{\mathbb \{R\}\})$-cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.},
author = {Avila, Artur, Bochi, Jairo},
journal = {Bulletin de la Société Mathématique de France},
keywords = {cocycle; minimal homeomorphism; uniform hyperbolicity; Lyapunov exponents},
language = {eng},
number = {3},
pages = {407-417},
publisher = {Société mathématique de France},
title = {A uniform dichotomy for generic $\{\rm SL\}(2,\{\mathbb \{R\}\})$ cocycles over a minimal base},
url = {http://eudml.org/doc/272427},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Avila, Artur
AU - Bochi, Jairo
TI - A uniform dichotomy for generic ${\rm SL}(2,{\mathbb {R}})$ cocycles over a minimal base
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 3
SP - 407
EP - 417
AB - We consider continuous ${\rm SL}(2,{\mathbb {R}})$-cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.
LA - eng
KW - cocycle; minimal homeomorphism; uniform hyperbolicity; Lyapunov exponents
UR - http://eudml.org/doc/272427
ER -

References

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  1. [1] J. Bochi – « Genericity of zero Lyapunov exponents », Ergodic Theory Dynam. Systems22 (2002), p. 1667–1696. Zbl1023.37006MR1944399
  2. [2] J. Bochi & M. Viana – « The Lyapunov exponents of generic volume-preserving and symplectic maps », Ann. of Math. (2) 161 (2005), p. 1423–1485. Zbl1101.37039MR2180404
  3. [3] S. J. Eigen & V. S. Prasad – « Multiple Rokhlin tower theorem: a simple proof », New York J. Math. 3A (1997/98), p. 11–14. Zbl0894.28009MR1604573
  4. [4] A. Furman – « On the multiplicative ergodic theorem for uniquely ergodic systems », Ann. Inst. H. Poincaré Probab. Statist.33 (1997), p. 797–815. Zbl0892.60011MR1484541
  5. [5] N. ĐìnhCống – « A generic bounded linear cocycle has simple Lyapunov spectrum », Ergodic Theory Dynam. Systems 25 (2005), p. 1775–1797. Zbl1130.37312MR2183293
  6. [6] J.-I. Nagata – Modern dimension theory, Bibliotheca Mathematica, Vol. VI. Edited with the cooperation of the “Mathematisch Centrum” and the “Wiskundig Genootschap” at Amsterdam, Interscience Publishers John Wiley & Sons, Inc., New York, 1965. Zbl0518.54002MR208571
  7. [7] J.-C. Yoccoz – « Some questions and remarks about SL ( 2 , 𝐑 ) cocycles », in Modern dynamical systems and applications, Cambridge Univ. Press, 2004, p. 447–458. Zbl1148.37306MR2093316

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