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There is no “Shadowing lemma” for partially hyperbolic dynamics. (Pas de “Shadowing Lemma” pour les dynamiques partiellement hyperboliques.) (French. Abridged English version) Zbl 0973.37016
From the text: The aim of this work is to understand if the Shadowing property can be verified, at least generically, for robustly transitive diffeomorphisms with a strong partially hyperbolic splitting. The answer to this question is definitively negative:
Theorem. Let $$f:M\to M$$ be a transitive diffeomorphism with a strong partially hyperbolic splitting defined on a compact 3-manifold. Assume that $$f$$ has two hyperbolic periodic points $$P$$ and $$Q$$ such that $$\dim(W^s (P))=2$$ and $$\dim(W^s (Q))=1$$. Then there is $$\varepsilon >0$$ such that, for all $$\delta >0$$, there is a $$\delta$$-pseudo-orbit which cannot be $$\varepsilon$$-shadowed by any orbit of $$f$$.
Corollary. There is a $$C^1$$-dense open subset $${\mathcal O}$$ of the set of non-Anosov, strong partially hyperbolic, and robustly transitive diffeomorphisms of a compact 3-manifold, such that every $$f\in{\mathcal O}$$ does not verify the shadowing property.
Let us observe that the proof of this result is really easy, however, it answers a question frequently posed by many collegues.

MSC:
 37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics 37D30 Partially hyperbolic systems and dominated splittings
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