Two-dimensional basic sets of structurally stable diffeomorphisms of three-dimensional manifolds (Q2759253)

The paper solves the problem of the non-existence of some classes of structurally stable diffeomorphisms on closed three-dimensional manifolds. Let \(f :M^n\to M^n\) be a structurally stable diffeomorphism of a closed \(n\)-dimensional manifold \(M^n\). Let \(\Omega\subset M^n\) be a codimension 1 basic set which for definiteness is assumed to be an attractor. The boundary of \(\Omega\) accessible from without is a union of finitely many unstable manifolds of periodic points of \(\Omega\) and can be decomposed into pairwise disjoint \(k\)-bunches, \(k\in\mathbb N\). It is known that if \(n\geq 3\) then \(\Omega\) contains only 1-bunches or 2-bunches. The main result is the following:NEWLINENEWLINENEWLINETheorem. Let \(f :M^3\to M^3\) be a structurally stable diffeomorphism of a closed three-dimensional manifold \(M^3\). Then \(f\) can have no two-dimensional basic sets with at least one 1-bunch.NEWLINENEWLINENEWLINEThe paper contains an outline of the proof.