On the typical diffeotopy of a structurally stable diffeomorphism with an expanding attractor of codimension one.
Two \(C^1\) diffeotopies \(\alpha(t)\) and \(\beta(t)\), \(t\in [0,1]\), in the space of diffeomorphisms on a smooth closed manifold are said to be {topologically stable} if there exists an orientation-preserving homeomorphism \(h\) on \([0,1]\) such that for any \(t\in [0,1]\), the diffeomorphisms \(\alpha(t)\) and \(\beta({h(t)})\) are topologically conjugate. A \(C^1\) diffeotopy \(\alpha\) is said to be {stable} if any nearby diffeotopy is topologically conjugate to \(\alpha\). A stable diffeotopy is said to be {simple} if it contains only finitely many diffeomorphisms which are not structurally stable. This paper states the result that if \(\dim(M)\geq 4\) and \(f\) is a structurally stable diffeomorphism on \(M\) with an orientable expanding attractor of codimension one, then there is a simple diffeotopy connecting \(f\) with some DA-diffeomorphism. It gives a partial answer to Problem 33 listed in [\textit{J. Palis} and \textit{C. C. Pugh}, Dyn. Syst., Proc. Symp. Univ. Warwick 1973/74, Lect. Notes Math. 468, 345--353 (1975; Zbl 0304.58011)].
scientific article; zbMATH DE number 2121058