Infinitesimal Lyapunov functions for singular flows

Details Infinitesimal Lyapunov functions for singular flows Infinitesimal Lyapunov functions for singular flows

2012-01-12

arxiv.org/abs/1201.2550v4

Mathematics

Dynamical Systems

39 pages, 1 figures; certain specific 3d results removed for another paper

Scientific paper

We present an extension of the notion of infinitesimal Lyapunov function to singular flows on three-dimensional manifolds, and show how this technique provides a characterization of partially hyperbolic structures for invariant sets for such flows, and also of sectional-hyperbolicity. In the absence of singularities, we can also rephrase uniform hyperbolicity with the language of infinitesimal Lyapunov functions. These conditions are expressed using the vector field X and its space derivative DX together with an infinitesimal Lyapunov function only and are reduced to checking that a certain symmetric operator is positive definite on the trapping region: we show how to express partial hyperbolicity using only the interplay between the infinitesimal generator X of the flow X_t, its derivative DX and the infinitesimal Lyapunov function.

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