On the inverse limit stability of endomorphisms
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 3, p. 463-475
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).
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).
@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},
     publisher = {Elsevier},
     volume = {30},
     number = {3},
     year = {2013},
     pages = {463-475},
     doi = {10.1016/j.anihpc.2012.10.001},
     zbl = {06295429},
     language = {en},
     url = {http://www.numdam.org/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, Volume 30 (2013) no. 3, pp. 463-475. doi : 10.1016/j.anihpc.2012.10.001. http://www.numdam.org/item/AIHPC_2013__30_3_463_0/

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