Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flows

Abstract

We prove that a volume-preserving three-dimensional flow can be C 1 -approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C 1 -topology.

Stability properties of divergence-free vector fields

“…On the divergence-free setting, Bessa and Rocha [30] considered a three-dimensional manifold M, proving the next result.…”

Dynamics of conservative Bykov cycles: Tangencies, generalized Cocoon bifurcations and elliptic solutions

“…Main results. By [6], the existence of hyperbolic equilibria implies that the flow is not C 1 -near an Anosov one and thus, C 1 -close to the flow, tangencies are expected. Furthermore, by [7], arbitrarily C 1 -close to a conservative flow displaying a Bykov cycle, we may find an incompressible flow with a dense set of elliptic periodic solutions.…”

“…The first author and Duarte proved in [7] that the set of C 1 -divergence-free vector fields defined in a compact three-dimensional Riemannian manifold without boundary, has a C 1 -residual set such that any vector field inside it is Anosov or else, the flow associated to it has dense elliptic solutions in the phase space. Furthermore, in [6], also in this context, the authors proved that if the vector field is not Anosov, then it can be C 1 -approximated by another divergence-free vector field exhibiting homoclinic tangencies. However, a conservative vector field whose flow has a persistent Bykov cycle may lie outside these residual/dense subsets and the developed theory cannot be applied for this degenerated class of systems.…”

Hamiltonian suspension of perturbed Poincaré sections and an application

“…In fact, any divergence-free vector field defined on a 3-dimensional closed manifold can be C 1 -approximated in the same class by a vector field either Anosov or with a homoclinic tangency associated to a hyperbolic closed orbit [4]. This was recently generalized in [9] for a d-dimensional closed manifold, d ≥ 4: any divergence-free vector field can be C 1 -approximated by another one satisfying either one of the properties of the 3-dimensional case, or with a heterodimensional cycle.…”

“…This was recently generalized in [9] for a d-dimensional closed manifold, d ≥ 4: any divergence-free vector field can be C 1 -approximated by another one satisfying either one of the properties of the 3-dimensional case, or with a heterodimensional cycle. Here we address the problem of obtaining a version of [4] in the Hamiltonian context.…”