Structural stability of attractor-repellor endomorphisms with singularities (Q2884072)

The paper deals with the so-called attractor-repeller endomorphisms of compact manifolds. By definition, this means that the non-wandering set is the disjoint union of two disjoint invariant compact sets: one of them, say \(R\), is expanding; the other one, say \(A\), is a hyperbolic invariant subset containing the unstable manifolds of its points, and the mapping is bijective on the latter subset. These invariant subsets are not necessarily transitive.NEWLINENEWLINEThe main result (Theorem 1.5) yields the following sufficient conditions for \(C^{\infty}\)-smooth structural stability of an attractor-repeller endomorphism \(f\):NEWLINENEWLINE(i) density of periodic points in \(A\);NEWLINENEWLINE(ii) the orbits of the singularities are disjoint from the non-wandering set;NEWLINENEWLINE(iii) \(C^{\infty}\)-infinitesimal stability of \(f\) on the interior of the complement to the backward orbit of the non-wandering set;NEWLINENEWLINE(iv) transversality of \(f\) to the stable manifolds of the points of \(A\).NEWLINENEWLINEThis is a very general result having important applications. It generalizes a well-known joint result of \textit{W. de Melo} and \textit{S. van Strien} [One-dimensional dynamics. Berlin: Springer (1993; Zbl 0791.58003)] which deals with attractor-repeller circle endomorphisms with quadratic critical points. Their result says that if the critical orbits are disjoint and non-preperiodic, then the endomorphism is \(C^{\infty}\) structurally stable. Theorem 1.5 also generalizes results of \textit{J. Iglesias, A. Portela} and \textit{A. Rovella} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 6, 1209--1220 (2008; Zbl 1153.37329); Fundam. Math. 208, No. 1, 23--33 (2010; Zbl 1196.37052)].NEWLINENEWLINEAnother result of the paper, Theorem 2.3 (used in the proof of Theorem 1.5), generalizes unpublished J. Mather's theorem from singularity theory, on the equivalent structural stability of graphs of composed mapping, to larger category of laminations. This result of Mather was written by \textit{N. A. Baas} in the first part of the preprint [``Structural stability of composed mappings'', Preprint (1974)].