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### Open Access Journals

DCDS

We study dynamical systems for which at most $n$ orbits can accompany
a given arbitrary orbit.
For simplicity we call them

*$n$-expansive*(or*positively $n$-expansive*if positive orbits are considered instead). We prove that these systems can satisfy properties of expansive systems or not. For instance, unlike positively expansive maps [3], positively $n$-expansive homeomorphisms may exist on certain infinite compact metric spaces. We also prove that a map (resp. bijective map) is positively $n$-expansive (resp. $n$-expansive) if and only if it is so outside finitely many points. Finally, we prove that a homeomorphism on a compact metric space is $n$-expansive if and only if it is so outside finitely many orbits. These last resuls extends previous ones for expansive systems [2],[11],[12].
DCDS

A

*singular-hyperbolic set*for flows is a partially hyperbolic set with singularities (hyperbolic ones) and volume expanding central direction [7]. Several properties of hyperbolic systems have been conjectured for the singular-hyperbolic sets [8, p. 335]. Related to these conjectures we shall prove the existence of transitive, isolated, singular-hyperbolic set*without periodic orbits*on any $3$-manifold. In particular, the periodic orbits are not necessarily dense in the limit set of a isolated singular-hyperbolic set.
DCDS

We prove that there is a residual subset $C$, in the space of all $\mathcal C^1$ vector fields
of a closed $n$-manifold $M$, such that for every $X \in \mathcal R$ the set of points in $M$ with Lyapunov
stable $\omega$-limit set is residual in $M$. This improves a result in Arnaud [1] and gives a partial
solution to a conjecture in Hurley [8].

DCDS

The

*sectional-hyperbolic sets*constitute a class of partially hyperbolic sets introduced in [20] to describe robustly transitive singular dynamics on $n$-manifolds (e.g. the multidimensional Lorenz attractor [9]). Here we prove that a transitive sectional-hyperbolic set with singularities contains no local strong stable manifold through any of its points. Hence a transitive, isolated, sectional-hyperbolic set containing a local strong stable manifold is a hyperbolic saddle-type repeller. In addition, a proper transitive sectional-hyperbolic set on a compact $n$-manifold has empty interior and topological dimension $\leq n-1$. It follows that a*singular-hyperbolic attractor*with singularities [22] on a compact $3$-manifold has topological dimension $2$. Hence such an attractor is*expanding*, i.e., its topological dimension coincides with the dimension of its central subbundle. These results apply to the robustly transitive sets considered in [22], [17] and also to the Lorenz attractor in the Lorenz equation [25].## Year of publication

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