Shadowing properties of \({\mathcal L}\)-hyperbolic homeomorphisms.

*(English)*Zbl 0983.37024The notion of \(\mathcal L\)-hyperbolic homeomorphisms on compact metric spaces is introduced as a generalization of Axiom A diffeomorphisms. It is shown to be equivalent to expansive homeomorphisms having the shadowing property and to Ruelle’s Smale spaces. For \(\mathcal L\)-hyperbolic homeomorphisms the Lipschitz shadowing property and average shadowing property are established. Some examples of \(\mathcal L\)-hyperbolic homeomorphisms are considered. Main results of the paper are the following.

Theorem 1. Let \(f\) be a homeomorphism of a compact metric space \(X\). The following conditions are equivalent:

(1) \(f\) is expansive and has the shadowing property,

(2) there is a compatible metric \(d\) for \(X\) such that \(f\) is \(\mathcal L\)-hyperbolic,

(3) \((X,f)\) is a Smale space.

Theorem 2. Let \(f:X\to X\) be a homeomorphism of a compact metric space \((X,d)\). If \(f\) is \(\mathcal L\)-hyperbolic then \(f\) has the Lipschitz shadowing property.

Theorem 3. Let \(f:(X,d)\to (X,d)\) be an \(\mathcal L\)-hyperbolic homeomorphism of a compact metric space \((X,d)\). The following conditions are equivalent:

(1) \(f\) has the average shadowing property,

(2) \(f\) is topologically transitive.

Theorem 1. Let \(f\) be a homeomorphism of a compact metric space \(X\). The following conditions are equivalent:

(1) \(f\) is expansive and has the shadowing property,

(2) there is a compatible metric \(d\) for \(X\) such that \(f\) is \(\mathcal L\)-hyperbolic,

(3) \((X,f)\) is a Smale space.

Theorem 2. Let \(f:X\to X\) be a homeomorphism of a compact metric space \((X,d)\). If \(f\) is \(\mathcal L\)-hyperbolic then \(f\) has the Lipschitz shadowing property.

Theorem 3. Let \(f:(X,d)\to (X,d)\) be an \(\mathcal L\)-hyperbolic homeomorphism of a compact metric space \((X,d)\). The following conditions are equivalent:

(1) \(f\) has the average shadowing property,

(2) \(f\) is topologically transitive.

Reviewer: Eugene Ershov (St.Peterburg)

##### MSC:

37C50 | Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |

54H20 | Topological dynamics (MSC2010) |

##### Keywords:

\(\mathcal L\)-hyperbolic homeomorphism; pseudo-orbit; shadowing property; Lipschitz shadowing property; average shadowing property; Axiom A diffeomorphism
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