\(C^ r\) Axiom A maps having strong transversality. (English) Zbl 0797.58044
Denote by \(R^ r(M)\) the set of all \(C^ r\) regular maps of a closed \(C^ \infty\) manifold \(M\). The author shows that the set of all \(f \in R^ r(M)\) (\(r \geq 1\)), satisfying Axiom A and strong transversality, is open in \(R^ r(M)\). This is a generalization of the result by Newhouse and Palis for \(C^ r\) diffeomorphisms.
Reviewer: Yu.E.Gliklikh (Voronezh)
MSC:
| 37C75 | Stability theory for smooth dynamical systems |
| 37D99 | Dynamical systems with hyperbolic behavior |
References:
| [1] | Hirsch, M.; Palis, J.; Pugh, C.; Shub, M., Neighborhoods of hyperbolic sets, Invent. Math., 9, 121-134 (1970) · Zbl 0191.21701 |
| [2] | Mańẽ, R.; Pugh, C., Stability of endomorphisms, (Warwick Dynamical Systems. Warwick Dynamical Systems, Lecture Notes in Mathematics, 468 (1975), Springer: Springer Berlin), 175-184 · Zbl 0315.58018 |
| [3] | Moriyasu, K., Inverse limit stability and transversality of \(C^1\) regular maps (1991), Tokyo Metropolitan University: Tokyo Metropolitan University Tokyo |
| [4] | Newhouse, S.; Palis, J., Cycles and bifurcation theory, Astérisque, 31, 44-140 (1976) · Zbl 0322.58009 |
| [5] | Palis, J.; de Melo, W., Geometric Theory of Dynamical Systems, An Introduction (1982), Springer: Springer Berlin · Zbl 0491.58001 |
| [6] | Przytycki, F., Anosov endomorphisms, Studia Math., 58, 249-285 (1976) · Zbl 0357.58010 |
| [7] | Przytycki, F., On \(Ω\)-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math., 60, 61-77 (1977) · Zbl 0343.58008 |
| [8] | Shub, M., Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91, 175-199 (1969) · Zbl 0201.56305 |
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