There is no “Shadowing lemma” for partially hyperbolic dynamics.
(Pas de “Shadowing Lemma” pour les dynamiques partiellement hyperboliques.)

*(French. Abridged English version)*Zbl 0973.37016From 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.

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.