Extending flows from isolated invariant sets

Abstract

AbstractLet M be a closed n-dimensional manifold, and let U be a n-dimensional isolating block such that U is smoothly embedded in M. Let φ be a smooth semi-flow on U and let Λ contained in U, be isolated and invariant under φ Then there exists a semi-flow φ′ on M which extends φ such that φ′ is Morse-Smale outside of U, and no new recurrence is introduced in U. The theorem is true for any finite number of pairwise-disjoint Ui. Furthermore, if Λ is hyperbolic, topologically transitive and is the closure of periodic orbits, then φ′ is an Axiom A flow and is Ω-stable. In dimensions two and three, we have the stronger result that φ′ is structurally stable. Also, as a corollary, we give sufficient conditions for the flow φ′ to be nonsingular. One application of the corollary permits the formation of allowable knots and links in three-manifolds such that there exists a structurally stable nonsingular Morse-Smale flow φ′ which contains the specified knots and links in Ω(φ′) Moreover, the knots and links can be specified to be any combination of attractors, repellers or saddles.