Symbolic Dynamics for Hyperbolic Flows

Let/, {t e R) be a differentiable flow on a compact manifold M. A compact in-variant set A containing no fixed points is called hyperbolic if the tangent bundle restricted to A can be written as the Whitney sum of three Zyj-invariant continuous subbundles TAM = E + Es + Eu, where Eis the one-dimensional bundle tangent to the flow, and there are constants c, X> 0 so that (a) ||l>/,(v)| | S cr * ||v| | for veE',t^0, (b) ||2>/-,(v) | g certo ||v| | for veE», t ^ 0. A hyperbolic set A is called basic if, (a) the periodic orbits of ft | A are dense in A, {b)ft\ is a topologically transitive flow, (c) there is an open set U 3 A with A = fi'e*/< ^ Basic hyperbolic sets occur in Smale's Axiom A flows [11], a class containing all known structurally stable flows. An important special case is an Anosov flow; here M itself is a hyperbolic set. We will outline a method for studying the structure of basic sets, namely sym-bolic dynamics. The space 2 „ = IL{1> •- » »} is compact when given the product topology (and {1, •••,«} the discrete topology). One writes x = (#,-)£>-< » for a point in 2n and x { = {x)t-. The shift homeomorphism a:2n-> 2n is defined by G{X)ì = Jfy+i. For A an « x n matrix of 0's and l's, the set 2A = {xe2„: AX/tXj+l = 1 for all i) is compact and tf-invariant. A basic hyperbolic set A will be closely related to a certain symbolic space IA.