February  2005, 13(2): 239-269. doi: 10.3934/dcds.2005.13.239

Robustly transitive singular sets via approach of an extended linear Poincaré flow

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China, China

2. 

School of Mathematic Sciences, Peking University, Beijing, 100871

Received  August 2004 Revised  December 2004 Published  April 2005

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 Poincaré flow "to singularities".
Citation: Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239
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