Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows.
Let \(M\) be a compact three-dimensional manifold (without boundary), and let \(X\) be a \(C^{1}\)-vector field on \(M\). It is proved that \(X\) can be approximated in the \(C^{1}\)-topology by a \(C^{1}\)-vector field~\(Y\) on~\(M\) showing one of the following properties: (1) \(Y\) is uniformly hyperbolic and satisfies the no-cycles condition; (2) \(Y\) has a homoclinic tangency; (3) \(Y\) has a singular cycle. This solves a conjecture of J.~Palis in this special case. For diffeomorphisms of a surface this conjecture has been solved by \textit{E. R. Pujals} and \textit{M. Sambarino} [Ann. Math. (2) 151, 961--1023 (2000; Zbl 0959.37040)]. Although the proofs are technical the paper is well written. It is well organized and the exposition is clear.
scientific article; zbMATH DE number 1975935
scientific article