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Diffeomorphisms with weak shadowing. (English) Zbl 0968.37009
The author studies the property of weak shadowing in dynamical systems introduced by R. M. Corless and the reviewer [J. Math. Anal. Appl. 189, 409-423 (1995; Zbl 0821.58036)]. He investigates the \(C^1\)-interior \(I\) of the set of diffeomorphisms of a \(C^{\infty}\) closed surface having the weak shadowing property.
It is shown that any system in \(I\) satisfies Axiom A and the no-cycle condition (due to the existence of \(C^1\)-open sets of topologically transitive non-Anosov diffeomorphisms on the three-torus, this result cannot be generalized to dimensions \(n>2\)). In addition, the nonwandering set of a diffeomorphism \(f\in I\) is finite if and only if \(f\) is a Morse-Smale diffeomorphism.

MSC:
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D15 Morse-Smale systems
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