#### DMCA

## Hyperbolic sets with nonempty interior, Discrete Contin

Venue: | Dyn. Syst |

Citations: | 4 - 1 self |

### Citations

1542 |
Hasselblatt: Introduction to the Modern Theory of Dynamical Systems
- Katok, B
- 1995
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Citation Context ...2. Theorem 4. (Shadowing Theorem) If Λ is a locally maximal hyperbolic set, then given any δ > 0 there exists an > 0 and η > 0 such that if {xj}j2j=j1 is an -chain for f with d(xj, Λ) < η, then there is a y which δ-shadows {xj}j 2 j=j1 . If the -chain is periodic, then y is periodic. If j2 = −j1 = ∞, then y is unique and y ∈ Λ. The Shadowing Theorem implies the following: Corollary 1. If Λ is a locally maximal hyperbolic set of a diffeomorphism f , then cl(Per(f |Λ)) = NW(f |Λ) = R(f |Λ). 4 TODD FISHER A standard result is the following Spectral Decomposition Theorem [4, p. 575]. (Note in [4] the result is stated for the nonwandering set, but from the above corollary this is equal to the chain recurrent set.) Theorem 5. (Spectral Decomposition) Let M be a Riemannian manifold, U ⊂ M open, f : U → M a diffeomorphic embedding, and Λ ⊂ U a compact locally maximal hyperbolic set for f . Then there exist disjoint closed sets Λ1, ..., Λm and a permutation σ of {1, ...,m} such that R(f |Λ) = ⋃m i=1 Λi, f(Λi) = Λσ(i), and when σ k(i) = i then fk|Λi is topologically mixing. A set X is topologically mixing for f provided that, for any open sets U and V in X, there is a positive integer n0 su... |

573 |
Differentiable dynamical systems
- Smale
- 1967
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Citation Context ...eorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal hyperbolic set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism. 1. Introduction For a diffeomorphism f of a closed connected manifold a hyperbolic set Λ is a compact f -invariant set whose tangent space splits into invariant uniformly contracting and uniformly expanding directions. On a compact manifold these sets often possess a very rich structure. The pioneering article by Smale [8] states many of the standard results for hyperbolic sets. Hyperbolic sets with nonempty interior are quite special. Indeed, we have: Theorem 1. Let f : M → M be a diffeomorphism of a compact manifold M . If f has a transitive hyperbolic set Λ with nonempty interior, then Λ = M and f is Anosov. Theorem 1 appears to be a well known folklore theorem. We could find no proof of it in the literature, so one is provided. Our second result shows that the hypothesis of transitivity in Theorem 1 can be replaced with local maximality and low dimensionality. We recall the definition of locally maximal hyp... |

243 | Dynamical systems. Stability, symbolic dynamics, and chaos, Second edition - Robinson - 1999 |

15 |
Hyperbolic nonwandering sets on two-dimensional manifolds
- Newhouse, Palis
- 1973
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Citation Context ... = x and h(Q) ∩ W u(Λa) = h((−1, 1)×F ) where F ⊂ (−1, 1). Furthermore, for each y ∈ F the set h((−1, 1)× {y}) is a neighborhood of h(0, y) in W u(h(0, y)). So for each point x ∈ Λa and N a sufficiently small neighborhood of x the set N ∩W u(Λa) is a lamination. A point x ∈ W u(Λa) is a u-border if it belongs to an arc h((−1, 1)× y), where y is an extreme point of a component of the complement of F in (−1, 1). Replacing stable with unstable we similarly define a point as an s-border. x W (x) W (x) s u Figure 1. A u-border point The following proposition follows from work of Palis and Newhouse [5] and stated explicitly in [2]. Proposition 3. A hyperbolic attractor Λa contains a u-border, but no s-border. If Λa does not possess any border, then f is Anosov. For x ∈ Λa an s-arch is a subset of W s(x) homeomorphic to a closed interval, such that the end points of α intersect Λa and no point in the interior of α intersects Λa. The following is shown in [2]. HYPERBOLIC SETS WITH NONEMPTY INTERIOR 7 Proposition 4. Let S be a compact surface. If Λ ⊂ S is a hyperbolic set containing a nontrivial hyperbolic attractor Λa and y ∈ Λ − Λa is contained in W s(Λa), then y is contained in an s-arch. T... |

6 |
Diffeomorphismes de Smales des surfaces.
- Bonatti, Langevin
- 1998
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Citation Context ... for any j ∈ {1, ..., l − 1} and any periodic point qj ∈ Λij , the point qj is heteroclinically related to p. We then have that Λij = Λi1 for all j. Hence, the relation as defined above has at most 1-cycles restricted to Λ. 2 The above theorem implies that we can talk of the lowest and highest elements in the relation . This will be useful in Section 3 in proving the existence of attractors and repellers. In the proof of Theorem 2 it will be useful to know more about the structure of hyperbolic attractors contained in compact smooth Riemannian surfaces. Most of this material is a review of [2]. A set Λa is a hyperbolic attractor provided Λa is a hyperbolic set, f |Λa is transitive, a neighborhood V of Λa exists such that f(cl(V )) ⊂ V , and Λa = ⋂ n∈N f n(V ). The neighborhood V is an attracting neighborhood for Λa. A hyperbolic attractor is nontrivial if it is not a periodic orbit. 6 TODD FISHER Similarly, a set Λr is a hyperbolic repeller provided Λr is a hyperbolic set, f |Λr is transitive, a neighborhood V of Λr exists such that f−1(cl(V )) ⊂ V , and Λr = ⋂ n∈N f −n(V ). A hyperbolic repeller is nontrivial if it is not a periodic orbit. The following standard result will be use... |

2 |
Finite Stability is not generic
- Robinson, Williams
- 1971
(Show Context)
Citation Context ...tting and has interior. The idea is to compactify the example. Specifically, take a diffeomorphism f of a compact surface M , such that M contains a hyperbolic repeller Λr containing a fixed point p and a hyperbolic attractor Λa containing a fixed point q where Λa ∩Λr = ∅ HYPERBOLIC SETS WITH NONEMPTY INTERIOR 13 and W u(p) t W s(q) 6= ∅. We show for a point z ∈ W u(p) t W s(q) and r sufficiently small that the set Λ = Λr ∪ Λa ∪ (⋃ n∈Z fn(Dr(z)) ) is a hyperbolic set with nonempty interior. Theorem 2 implies that Λ is not contained in a locally maximal hyperbolic set since f is not Anosov. In [7] a diffeomorphism g is constructed on a compact surface of genus two containing a DA-attractor, Λa, and DA-repeller, Λr, such that int(W s(Λa) ∩W u(Λr)) 6= ∅. Pick periodic points p ∈ Λr and q ∈ Λa. Since Λa and Λr are topologically mixing locally maximal sets with periodic points dense, we have that W s(q) = W s(Λa) and W u(p) = W u(Λr). Hence, (W s(q) t W u(p)) ∩ int(W s(Λa) ∩W u(Λr) ∩ Λ) 6= ∅. Fix n ∈ N such that p and q are fixed under gn, let f = gn, and fix z ∈ W u(p) t W s(q). The first step is to define a continuous invariant splitting for Λ. If r, > 0 are sufficiently small, then fo... |