The topology and dynamics of flows

Abstract

Abstract. After a brief survey of various types of flows (Morse-Smale, Smale, Anosov, & partially hyperbolic) we focus on Smale flows on S 3 . However, we do give some consideration to Smale flows on other three-manifolds and to Smale diffeomorphisms. FlowsLet M be a compact connected Riemannian manifold without boundary. Let || · || be the norm on the tangent bundle T M and d(·, ·) the metric induced on M . By a flow on M we mean a smooth function f :Much of what we describe in this sections for flows carries over with suitable modifications to diffeomorphisms.The chain recurrent set of a flow f isThe chain recurrent set of a flow is said to have a hyperbolic structure if the tangent bundle of the manifold structure can be written as a Whitney sums of sub-bundles invariant under Df where E c x is the subspace of T M x corresponding to the orbit of x and such that there are constants C > 0 and λ > 0 for which ||Df t (v)|| ≤ Ce −λt ||x|| for v ∈ E s , t ≥ 0 and ||Df t (v)|| ≥ 1/Ce λt ||x|| for v ∈ E u , t ≥ 0. Steve Smale showed that when R is hyperbolic it is the closure of the periodic orbits of the flow. Smale also showed that when R is hyperbolic it has a finite decomposition into compact invariant sets called basic sets:Theorem (Spectral Decomposition Theorem). Suppose that the chain recurrent set R of a flow has a hyperbolic structure. Then R is a finite disjoint union of compact invariant sets Λ 1 , Λ 2 , . . . , Λ k where each Λ i contains an orbit that is dense in Λ i .We define respectively the stable and unstable manifolds of an orbit O in a flow f . W s (O) = {y ∈ M | d(f (y, t), f (x, t)) → 0 as t → ∞ for some x ∈ O} W u (O) = {y ∈ M | d(f (y, t), f (x, t)) → 0 as t → −∞ for some x ∈ O}