Codimension one structurally stable chain classes (Q2790610)

The authors provide an interesting result regarding chain classes. More precisely, they generalize a theorem proved by Palis and Smale, that states that each diffeomorphism in a finite dimensional compact manifold has only hyperbolic critical elements. Such results were also explored in [\textit{J. K. Hale} et al., An introduction to infinite dimensional dynamical systems -- geometric theory. With an appendix by Krzysztof P. Rybakowski. New York etc.: Springer-Verlag (1984; Zbl 0533.58001)] in their book of infinite dimensional dynamical systems.NEWLINENEWLINENow, the authors work with \textit{chain classes}, which are equivalence classes of \(CR(f)\) -- the set of chain recurrent points of a diffeomorphism \(f\) -- under the relation of existence of pseudo-orbits of any small size. Each chain class is compact and invariant under \(f\), and cannot be decomposed in smaller compact invariant sets, so these classes represent the smallest invariant sets of \(f\) in \(CR(f)\). To study each one individually is an important and hard task to achieve.NEWLINENEWLINEHere, they prove the analogous result for codimension 1, that is, if \(f\) is a diffeomorphism of index \(1\) or \(\text{dim}\,M-1\), in a finite dimensional compact manifold \(M\) with a hyperbolic periodic point \(p\) such that its chain class is structurally stable, then it is also hyperbolic.NEWLINENEWLINEWe note that this study is absolutely non-trivial and the proofs do not follow as simple generalizations, since there is no assumption regarding the maximality of the chain classes.