Volume 8, Issue 1

We identify using the rational S 1 -cohomology of the space C β the odd generators of the homology for contact forms defined in [3].

We extend in this paper the definition of the homology for contact forms defined in [1] to the case when the ”dual” form β is not a contact form anymore.

In this paper we prove that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential. The technique is quantitative: as we introduce a large amount of rotation near the zeroes of the vector field in the kernel, we track down the modification of the variational problem and provide bounds on a key quantity (denoted by τ).

In this paper we study a functional at infinity associated to a contact form on a three dimensional manifold. The Morse index of this functional at infinity can be decomposed into two parts, one along the characteristic manifold and the other along the normal directions. We prove that we can redistribute the negative directions between the two subspaces through a local deformation of the contact form near a critical point at infinity.

In this note we consider the problem where u, ϕ ∈ H 0 1 (Ω) and Ω ⊂ ℝ 3 is a smooth bounded domain. We are interested in existence and multiplicity of solutions depending on the parameters p and λ. Our results extend previous work made in ℝ 3 .

We prove that there exists a residual subset R (with respect to the C 0 topology) of d-dimensional linear differential systems based in a μ-invariant flow and with transition matrix evolving in GL(d,ℝ) such that if A ∈ R, then, for μ-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial. Moreover, in the conservative setting, we obtain the dichotomy: dominated splitting versus zero Lyapunov exponents.