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).
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).
@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}, pages = {463--475}, publisher = {Elsevier}, volume = {30}, number = {3}, year = {2013}, doi = {10.1016/j.anihpc.2012.10.001}, zbl = {06295429}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2012.10.001/} }
TY - JOUR AU - Berger, Pierre AU - Rovella, Alvaro TI - On the inverse limit stability of endomorphisms JO - Annales de l'I.H.P. Analyse non linéaire PY - 2013 SP - 463 EP - 475 VL - 30 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2012.10.001/ DO - 10.1016/j.anihpc.2012.10.001 LA - en ID - AIHPC_2013__30_3_463_0 ER -
%0 Journal Article %A Berger, Pierre %A Rovella, Alvaro %T On the inverse limit stability of endomorphisms %J Annales de l'I.H.P. Analyse non linéaire %D 2013 %P 463-475 %V 30 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2012.10.001/ %R 10.1016/j.anihpc.2012.10.001 %G en %F 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, Volume 30 (2013) no. 3, pp. 463-475. doi : 10.1016/j.anihpc.2012.10.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.10.001/
[1] -maps having hyperbolic periodic points, Fund. Math. 169 no. 1 (2001), 1-49 | EuDML | Zbl
, , ,[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
,[4] Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397 | Zbl
,[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
, ,[6] stable maps: examples without saddles, Fund. Math. 208 no. 1 (2010), 23-33 | EuDML | Zbl
, , ,[7] Density of hyperbolicity in dimension one, Ann. of Math. (2) 166 no. 1 (2007), 145-182 | Zbl
, , ,[8] Laminations in holomorphic dynamics, J. Differential Geom. 47 no. 1 (1997), 17-94 | Zbl
, ,[9] A proof of the stability conjecture, Inst. Hautes Études Sci. Publ. Math. no. 66 (1988), 161-210 | EuDML | Numdam | Zbl
,[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 | EuDML | Zbl
,[12] On Ω-stability and structural stability of endomorphisms satisfying axiom A, Studia Math. 60 no. 1 (1977), 61-77 | EuDML | Zbl
,[13] Stability of Anosov maps, Proc. Amer. Math. Soc. 104 no. 1 (1988), 303-309 | Zbl
,[14] Structural stability of diffeomorphisms, J. Differential Equations 22 no. 1 (1976), 28-73 | Zbl
,[15] Stabilité globale des systèmes dynamiques, Astérisque vol. 56, Société Mathématique de France, Paris (1978) | Numdam | Zbl
,Cited by Sources: