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Nonexpanding attractors: Conjugacy to ALGEBRAIC MODELS AND CLASSIFICATION IN 3manifolds
, 2010
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Constraints On Dynamics Preserving Certain Hyperbolic Sets
, 2009
"... We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First ifΛis a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, thenΛis a union of transitive, expanding attractors (or contracting repellers) of codimension ..."
Abstract

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We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First ifΛis a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, thenΛis a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such thatΛis hyperbolic. Secondly, ifΛis a nonwandering, locally maximal, compact hyperbolic set for a surface diffeomorphism, thenΛis locally maximal for any diffeomorphism for whichΛis hyperbolic.
TRANSITIVE HYPERBOLIC SETS ON SURFACES
"... Abstract. We show that every transitive hyperbolic set on a surface is included in a locally maximal hyperbolic set. 1. History The history of hyperbolic dynamics can be traced back to two related directions of research: First, the study of geodesic flows such as the work of Hadamard, Hedlund, and ..."
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Abstract. We show that every transitive hyperbolic set on a surface is included in a locally maximal hyperbolic set. 1. History The history of hyperbolic dynamics can be traced back to two related directions of research: First, the study of geodesic flows such as the work of Hadamard, Hedlund, and Hopf. Second, homoclinic tangles and celestial mechanics starting with the work of Poincare ́ and continued, for instance, by Cartwright, Littlewood, and Smale. (Additionally, 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 finvariant 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 phenomena, such as, positive entropy and the existence of topological and measuretheoretic 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.