TRANSITIVE HYPERBOLIC SETS ON SURFACES

Abstract. We show that every transitive hyperbolic set on a sur-face is included in a locally maximal hyperbolic set. 1. History The history of hyperbolic dynamics can be traced back to two re-lated directions of research: First, the study of geodesic flows such as the work of Hadamard, Hedlund, and Hopf. Second, homoclinic tan-gles and celestial mechanics starting with the work of Poincare ́ and continued, for instance, by Cartwright, Littlewood, and Smale. (Ad-ditionally, Smale [10] adds a third direction of structural stability as studied by Andronov, Pontryagin, Lefschetz, Peixoto, and others.) If M is a compact, smooth, boundaryless manifold and f: M →M is a diffeomorphism, then a compact invariant set Λ is hyperbolic if the tangent space of Λ splits into f-invariant subbundles Es⊕Eu such that Es is uniformly exponentially contracted by the derivative of f, denoted Df, and Eu is uniformly exponentially expanded by Df. We note that hyperbolicity leads to many other interesting phenom-ena, such as, positive entropy and the existence of topological and measure-theoretic Markov models. In addition, for a point x if we define the stable and unstable sets, respectively, as follows: W s(x) = n≥0 f −n (W s (f n(x))) , and W u(x) = n≥0 f n (W u (f −n(x))) where W s (x) = {y ∈M | d(fn(x), fn(y)) ≤ for all n ≥ 0} and W u (x) = {y ∈M | d(fn(x), fn(y)) ≤ for all n ≤ 0}, then a standard result states that if f: M → M is a diffeomorphism, Λ is a hyperbolic set for f, and x ∈ Λ, then W s(x) and W u(x) are injectively immersed submanifolds