C1-stably shadowable conservative diffeomorphisms are Anosov

In this short note we prove that if a symplectomorphism f is C^1-stably shadowable, then f is Anosov. The same result is obtained for volume-preserving diffeomorphisms.info:eu-repo/semantics/publishedVersio

Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

Let X^0,1( M ) be the subset of divergence-free Lipschitz vector fields defined on a closed Riemannian manifold M endowed with the Lipschitz topology ∥ · ∥_0,1 where ν is the volume measure. Let L be the subset of vector fields satisfying the L-property, a property that implies C^1-regularity ν-almost everywhere. We prove that there exists a residual subset R ⊂ L with respect to ∥·∥0,1 such that Pesin’s entropy formula holds, i.e. for any X ∈ R the metric entropy equals the integral, with respect to ν, of the sum of the positive Lyapunov exponents.info:eu-repo/semantics/publishedVersio

Plenty of hyperbolicity on a class of linear homogeneous jerk differential equations

We consider 3×3 partially hyperbolic linear differential systems over an ergodic flow X^t and derived from the linear homogeneous differential equation x''(t)+β(X^t(t))x'(t)+ γ(t)x(t) = 0. Assuming that the partial hyperbolic decomposition E^s ⊕ E^c ⊕ E^u is proper and displays a zero Lyapunov exponent along the central direction E^c we prove that some C^0 perturbation of the parameters β(t) and γ(t) can be done in order to obtain non-zero Lyapunov exponents and so a chaotic behaviour of the solution.info:eu-repo/semantics/publishedVersio

Tracing orbits on conservative maps

We explore uniform hyperbolicity and its relation with the pseudo orbit tracing property. This property indicates that a sequence of points which is nearly an orbit (affected with a certain error) may be shadowed by a true orbit of the system. We obtain that, when a conservative map has the shadowing property and, moreover, all the conservative maps in a C1-small neighborhood display the same property, then the map is globally hyperbolicinfo:eu-repo/semantics/publishedVersio

Dynamics of generic multidimensional linear differential systems

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.info:eu-repo/semantics/publishedVersio

On C1-generic chaotic systems in three-manifolds

Let M be a closed 3-dimensional Riemannian manifold. We exhibit a C1-residual subset of the set of volume-preserving 3-dimensional flows defined on very general manifolds M such that, any flow in this residual has zero metric entropy, has zero Lyapunov exponents and, nevertheless, is strongly chaotic in Devaney’s sense. Moreover, we also prove a corresponding version for the discrete-time case.info:eu-repo/semantics/publishedVersio

Dynamics of generic 2-dimensional linear differential systems

We prove that for a C0-generic (a dense Gδ) subset of all the 2-dimensional conservative nonautonomous linear differential systems, either Lyapunov exponents are zero or there is a dominated splitting μ almost every point.info:eu-repo/semantics/publishedVersio

Hamiltonian elliptic dynamics on symplectic 4-manifolds

We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this implies that for far from Anosov regular energy surfaces of a C2-generic Hamiltonian the elliptic closed orbits are generic.Comment: 9 page

Removing zero Lyapunov exponents in volume-preserving flows

Baraviera and Bonatti proved that it is possible to perturb, in the c^1 topology, a volume-preserving and partial hyperbolic diffeomorphism in order to obtain a non-zero sum of all the Lyapunov exponents in the central direction. In this article we obtain the analogous result for volume-preserving flows.Comment: 10 page

On the fundamental regions of a fixed point free conservative Hénon map

It is well known that an orientation-preserving homeomorphism of the plane without fixed points has trivial dynamics; that is, its non-wandering set is empty and all the orbits diverge to infinity. However, orbits can diverge to infinity in many different ways (or not) giving rise to fundamental regions of divergence. Such a map is topologically equivalent to a plane translation if and only if it has only one fundamental region. We consider the conservative, orientation-preserving and fixed point free Hénon map and prove that it has only one fundamental region of divergence. Actually, we prove that there exists an area-preserving homeomorphism of the plane that conjugates this Hénon map to a translation.The first author was supported by FCT-FSE, SFRH/BPD/20890/2004. The second author was partially supported by FCT-POCTI/MAT/61237/2004info:eu-repo/semantics/publishedVersio