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Contributions to the geometric and ergodic theory of conservative flows

Published online by Cambridge University Press:  22 August 2012

MÁRIO BESSA  and
JORGE ROCHA
MÁRIO BESSA
Affiliation:
Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama, 6201-001 Covilhã, Portugal (email: bessa@fc.up.pt)
JORGE ROCHA
Affiliation:
Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal (email: jrocha@fc.up.pt)
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Abstract

We prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1709 - 1731
Copyright
Copyright © 2012 Cambridge University Press 

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