We prove that for any s 0 the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-∞. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.
We show that the integrated Lyapunov exponents of C 1 volume-preserving diﬀeomorphisms are simultaneously continuous at a given diﬀeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents are equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume-preserving diﬀeomorphisms on any compact manifold:
In this paper we consider the upper (lower) - stability of Lyapunov exponents of linear differential equations in Rn . Sufficient conditions for the upper - stability of maximal exponent of linear systems under linear perturbations are given. The obtained results are extended to the system with nonlinear perturbations.
Inspired by Lorenz’ remarkable chaotic ﬂow, we describe in this paper the structure of all C 1 robust transitive sets with singularities for ﬂows on closed 3-manifolds: they are partially hyperbolic with volume-expanding central direction, and are either attractors or repellers. In particular, any C 1 robust attractor with singularities for ﬂows on closed 3-manifolds always has an invariant foliation whose leaves are forward contracted by the ﬂow, and has positive Lyapunov exponent at every orbit, showing that any C 1 robust attractor resembles a geometric Lorenz attractor. ...