Abstract
\(C^r(B,B)\) is the space of \(C^r\) maps of a Banach manifold B to itself, bounded together with their derivatives up to the order r. It is chosen a topological subspace \(KC^r (B,B)\) satisfying certain compactness and reversibility conditions, the subspace depending on the class of problems in view. A large class of maps in \(KC^r(B,B),\) called Morse-Smale maps and the notion of stability relative to the largest invariant set A(f) are defined, and it is proved that the Morse-Smale maps are stable relatively to A(f) and form an open set in \(KC^r (B,B), r\ge 1.\) Examples of \(KC^r (B,B)\) can be constructed with maps arising from flows of retarded functional differential equations, of certain types of neutral functional differential equations and parabolic PDE and some other special PDE. Also, if B is compact, the set of all \(C^r-\)diffeomorphisms of B is a particular example of \(KC^r (B,B)\) and the main result yields the proof for the stability of Morse–Smale diffeomorphisms of a compact manifold, originally established in Palis (Topology 8:385–405, 1969) and Palis and Smale (in: Global analysis proc symp pure math, vol 14, AMS, Providence, RI, 1970).
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Acknowledgement
The author visited IMPA during the preparation of the present paper and would like to thank J. Palis and D. Henry for valuable suggestions and useful comments. IMPA, December 1982. This research has been supported in part by CNPq, Processo no. 102532-81.
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Communicated by Clodoaldo Grotta Ragazzo.
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Oliva, W.M. Stability of Morse-Smale maps. São Paulo J. Math. Sci. 16, 282–313 (2022). https://doi.org/10.1007/s40863-022-00294-z
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DOI: https://doi.org/10.1007/s40863-022-00294-z