Robustly transitive singular sets via approach of an extended linear Poincaré flow
By Ming Li, Shaobo Gan, Lan Wen. 2005
82 citations recorded by the source corpus.
Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a 3-dimensional flow must be partially hyperbolic. In this paper we generalize the result to higher dimensions. By definition, an isolated invariant set $\Lambda$ of a $C^1$ vector field $S$ is called robustly transitive if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that for every $X\in\mathcal U$, the maximal invariant set of $X$ in $U$ is non-trivially transitive. Such a set $\Lambda$ is called singular if it contains a singularity. The set $\Lambda$ is called strongly homogeneous of index $i$, if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that all periodic orbits of all $X\in\mathcal U$ contained in $U$ have the same index $i$. We prove in this paper that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits fit in certain way. As corollaries we obtain that every robust singular attractor (or repeller) that is strongly homogeneous must be partially hyperbolic and, if dim$M\le 4$, every robustly transitive singular set that is strongly homogeneous must be partially hyperbolic. The main novelty of the proofs in this paper is an extension of the usual linear Poincare flow "to singularities".
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/7dee5d23166f5fe9a2120e79e13a4eaebe7686f4
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Citations in the corpus
- Morse-Smale systems and horseshoes for three dimensional singular flows (2013, background): “Ñ is a (d − 1)-dimensional vector bundle on the base space Λ̃ (for a formal discussion, see [24]).”