Nous présentons différents résultats suggérant que le concept de -stabilité (structurelle de la limite inverse) est indépendant de la théorie des singularités. Nous décrivons un exemple dʼun endomorphisme robustement transitif et -stable ayant un ensemble critique persistant. Nous montrons que tout endomorphisme axiome A et -stable vérifie nécessairement une certaine condition de transversalité forte (T). Nous démontrons que tout endomorphisme attracteur–répulseur vérifiant la condition (T) est -stable. Ce dernier résultat est appliqué, entre autres, aux applications de type Hénon et aux fractions rationnelles. Cela nous amène à conjecturer que les endomorphismes -stables sont exactement ceux qui vérifient lʼaxiome A et la condition (T).
We present several results suggesting that the concept of -inverse (limit structural) stability is free of singularity theory. An example of a robustly transitive, -inverse stable endomorphism with a persistent critical set is given. We show that every -inverse stable, axiom A endomorphism satisfies a certain strong transversality condition (T). We prove that every attractor–repeller endomorphism satisfying axiom A and condition (T) is -inverse stable. The latter is applied to Hénon maps, rational functions and others. This leads us to conjecture that -inverse stable endomorphisms are exactly those which satisfy axiom A and condition (T).
@article{AIHPC_2013__30_3_463_0,
author = {Berger, Pierre and Rovella, Alvaro},
title = {On the inverse limit stability of endomorphisms},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
volume = {30},
year = {2013},
pages = {463-475},
doi = {10.1016/j.anihpc.2012.10.001},
zbl = {06295429},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_3_463_0}
}
Berger, Pierre; Rovella, Alvaro. On the inverse limit stability of endomorphisms. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 463-475. doi : 10.1016/j.anihpc.2012.10.001. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_3_463_0/
[1] , , , -maps having hyperbolic periodic points, Fund. Math. 169 no. 1 (2001), 1-49 | Zbl 1031.37021
[2] P. Berger, Persistence of stratification of normally expanded laminations, Mem. Soc. Math. Fr. (N.S.), in press, arXiv:math.DS.
[3] , Persistence of laminations, Bull. Braz. Math. Soc. 41 no. 2 (2008), 259-6319 | Zbl 1225.37037
[4] , Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397 | Zbl 0212.29103
[5] , , Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, Real and Complex Dynamical Systems, Hillerød, 1993, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. vol. 464, Kluwer Acad. Publ., Dordrecht (1995), 89-132 | Zbl 0874.54037
[6] , , , stable maps: examples without saddles, Fund. Math. 208 no. 1 (2010), 23-33 | Zbl 1196.37052
[7] , , , Density of hyperbolicity in dimension one, Ann. of Math. (2) 166 no. 1 (2007), 145-182 | Zbl 1138.37013
[8] , , Laminations in holomorphic dynamics, J. Differential Geom. 47 no. 1 (1997), 17-94 | Zbl 0910.58032
[9] , A proof of the stability conjecture, Inst. Hautes Études Sci. Publ. Math. no. 66 (1988), 161-210 | Numdam | Zbl 0678.58022
[10] , , Stability of endomorphisms, Dynamical Systems, Warwick, 1974, Lecture Notes in Math. vol. 468, Springer, Berlin (1975), 175-184
[11] , Anosov endomorphisms, Studia Math. 58 no. 3 (1976), 249-285 | Zbl 0357.58010
[12] , On Ω-stability and structural stability of endomorphisms satisfying axiom A, Studia Math. 60 no. 1 (1977), 61-77 | Zbl 0343.58008
[13] , Stability of Anosov maps, Proc. Amer. Math. Soc. 104 no. 1 (1988), 303-309 | Zbl 0664.58019
[14] , Structural stability of diffeomorphisms, J. Differential Equations 22 no. 1 (1976), 28-73 | Zbl 0343.58009
[15] , Stabilité globale des systèmes dynamiques, Astérisque vol. 56, Société Mathématique de France, Paris (1978) | Zbl 0396.58014