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Periodic orbits and chain-transitive sets of \(C^1\)-diffeomorphisms. (English) Zbl 1226.37007
Summary: We prove that the chain-transitive sets of \(C^1\)-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes.
This result is a consequence of a global connecting lemma, which allows to build by a \(C^1\)-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by \(C^1\)-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in \(\text{Diff}^1(M)\).

MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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