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Shadowing properties of $${\mathcal L}$$-hyperbolic homeomorphisms. (English) Zbl 0983.37024
The 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.

##### 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)
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