no. 3
On the inverse limit stability of endomorphisms
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 3, pp. 463-475.

Nous présentons différents résultats suggérant que le concept de C 1 -stabilité (structurelle de la limite inverse) est indépendant de la théorie des singularités. Nous décrivons un exemple dʼun endomorphisme robustement transitif et C 1 -stable ayant un ensemble critique persistant. Nous montrons que tout endomorphisme axiome A et C 1 -stable vérifie nécessairement une certaine condition de transversalité forte (T). Nous démontrons que tout endomorphisme attracteur–répulseur vérifiant la condition (T) est C 1 -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 C 1 -stables sont exactement ceux qui vérifient lʼaxiome A et la condition (T).

We present several results suggesting that the concept of C 1 -inverse (limit structural) stability is free of singularity theory. An example of a robustly transitive, C 1 -inverse stable endomorphism with a persistent critical set is given. We show that every C 1 -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 C 1 -inverse stable. The latter is applied to Hénon maps, rational functions and others. This leads us to conjecture that C 1 -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},
     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://www.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://www.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://www.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, Tome 30 (2013) no. 3, pp. 463-475. doi : 10.1016/j.anihpc.2012.10.001. http://www.numdam.org./articles/10.1016/j.anihpc.2012.10.001/

[1] N. Aoki, K. Moriyasu, N. Sumi, C 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] P. Berger, Persistence of laminations, Bull. Braz. Math. Soc. 41 no. 2 (2008), 259-6319 | Zbl

[4] Rufus Bowen, Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377-397 | Zbl

[5] J.H. Hubbard, R.W. Oberste-Vorth, 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] J. Iglesias, A. Portela, A. Rovella, C 1 stable maps: examples without saddles, Fund. Math. 208 no. 1 (2010), 23-33 | EuDML | Zbl

[7] O. Kozlovski, W. Shen, S. Van Strien, Density of hyperbolicity in dimension one, Ann. of Math. (2) 166 no. 1 (2007), 145-182 | Zbl

[8] M. Lyubich, Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 no. 1 (1997), 17-94 | Zbl

[9] R. Mañé, A proof of the C 1 stability conjecture, Inst. Hautes Études Sci. Publ. Math. no. 66 (1988), 161-210 | EuDML | Numdam | Zbl

[10] R. Mañé, C. Pugh, Stability of endomorphisms, Dynamical Systems, Warwick, 1974, Lecture Notes in Math. vol. 468, Springer, Berlin (1975), 175-184

[11] Feliks Przytycki, Anosov endomorphisms, Studia Math. 58 no. 3 (1976), 249-285 | EuDML | Zbl

[12] F. Przytycki, On Ω-stability and structural stability of endomorphisms satisfying axiom A, Studia Math. 60 no. 1 (1977), 61-77 | EuDML | Zbl

[13] J. Quandt, Stability of Anosov maps, Proc. Amer. Math. Soc. 104 no. 1 (1988), 303-309 | Zbl

[14] C. Robinson, Structural stability of C 1 diffeomorphisms, J. Differential Equations 22 no. 1 (1976), 28-73 | Zbl

[15] M. Shub, Stabilité globale des systèmes dynamiques, Astérisque vol. 56, Société Mathématique de France, Paris (1978) | Numdam | Zbl

Cité par Sources :