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The ergodic shadowing property from the robust and generic view point

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Lee Advances in Difference Equations 2014, 2014:170 http://www.advancesindifferenceequations.com/content/2014/1/170 RESEARCH Open Access The ergodic shadowing property from the robust and generic view point Manseob Lee* * Correspondence: lmsds@mokwon.ac.kr Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea Abstract In this paper, we discuss that if a diffeomorphisms has the C 1 -stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, C 1 -generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set. MSC: 34D30; 37C20 Keywords: ergodic shadowing; shadowing; locally maximal; generic; Anosov 1 Introduction Let M be a closed C ∞ manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C  -topology. Denote by d the distance on M induced from a Riemannian metric  ·  on the tangent bundle TM. Let f ∈ Diff(M). For δ > , a sequence of points {xi }bi=a (–∞ ≤ a < b ≤ ∞) in M is called a δ-pseudo orbit of f if d(f (xi ), xi+ ) < δ for all a ≤ i ≤ b – . For given x, y ∈ M, we write x  y if for any δ > , there is a δ-pseudo orbit {xi }bi=a (a < b) of f such that xa = x and xb = y. Let  be a closed f -invariant set. We say that f has the shadowing property in  if for every  >  there is δ >  such that, for any δ-pseudo orbit {xi }bi=a ⊂  of f (–∞ ≤ a < b ≤ ...
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