Anosov diffeomorphisms constructed from \(\pi _{k}\)(Diff\((S^{n}))\) (Q2892841)

This article investigates a general question about the relationship between dynamics and topology, namely finding conditions (or obstructions) for the existence, on certain manifolds, of diffeomorphisms or flows with particular dynamical properties.NEWLINENEWLINEThe dynamical systems under consideration in this article are Anosov diffeomorphisms. These are diffeomorphisms \(f:M\rightarrow M\) of a smooth Riemannian manifold \(M\), for which the tangent bundle \(TM\) admits a \(df\)-invariant splitting \(TM=E^s\oplus E^u\) into stable and unstable subbundles \(E^s\) and \(E^u\), i.e., for all non-negative integer \(m\), one has NEWLINE\[NEWLINE \begin{aligned} \|df^m v\|&\leqslant C\lambda^m \| v\|,\quad v\in E^s,\\ \|df^{-m} v\|&\leqslant C\lambda^m \| v\|,\quad v\in E^u, \end{aligned} NEWLINE\]NEWLINE where \(C>0\) and \(\lambda\in(0,1)\) are some constants.NEWLINENEWLINEAnosov diffeomorphisms on compact manifolds are interesting from the dynamical point of view since they provide simple examples of chaotic systems (like, e.g., certain torus automorphisms). On the other hand, the class of compact manifolds admitting an Anosov diffeomorphism is quite restricted, namely it is conjectured to be the class of infranilmanifolds.NEWLINENEWLINEA contribution toward the classification of Anosov diffeomorphisms is given in this paper by studying the existence of Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds but with an exotic smooth structure. Making use of connected sums of standard infranilmanifolds with certain exotic spheres, they construct Anosov diffeomorphisms of higher codimension in comparison with a previous work of one of the authors [\textit{F. T. Farrell} and \textit{L. E. Jones}, Topology 17, 273--282 (1978; Zbl 0407.58031)]. Here, an Anosov diffeomorphism \(f\) is said to be of codimension \(k\), if the fiber of \(E^s\) (or \(E^u\)) has dimension \(k\), with \(k\leqslant \lfloor n/2 \rfloor\), where \(n\) is the dimension of \(M\).