Contributions to the Geometric and Ergodic Theory of Conservative Flows

Details Contributions to the Geometric and Ergodic Theory of Conservative Flows Contributions to the Geometric and Ergodic Theory of Conservative Flows

2008-10-21

arxiv.org/abs/0810.3855v1

Mathematics

Dynamical Systems

26 pages, 2 figures

Scientific paper

We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

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