On the inverse limit stability of endomorphisms

Berger, Pierre; Rovella, Alvaro

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).

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