The explosion of singular-hyperbolic attractors

Abstract

A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if $\Lambda$ is a singular-hyperbolic attractor of a three-dimensional vector field X, then there is a neighborhood U of $\Lambda$ in M such that every attractor in U of a C^r vector field C^r close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of $\Lambda$ (in M) and $\mathcal U$ of X (in the space of C^r vector fields) such that if n denotes the number of singularities of X in $\Lambda$, then $\#\{A\subset U:A$ is an attractor of $Y\in\mathcal U\}\leq n$. Every three-dimensional vector field C^r close to one exhibiting a singular-hyperbolic attractor has a singularity non-isolated in the non-wandering set. A singularity of a three-dimensional C^r vector field Y is stably non-isolated in the non-wandering set if it is the unique singularity of a singular-hyperbolic attractor of Y. These results generalize well-known properties of the geometric Lorenz attractor.