Tame dynamics and robust transitivity

TL;DR

This paper investigates the fragility of chain-recurrence classes in smooth dynamical systems, constructing examples of tame systems with non-transitive classes, highlighting limitations of generic properties.

Contribution

It constructs a C1-open set of tame diffeomorphisms with a dense subset where some chain-recurrence classes are non-transitive, revealing the fragility of generic properties.

Findings

Existence of tame diffeomorphisms with non-transitive chain-recurrence classes.

Demonstration that the property of classes being homoclinic is fragile.

Construction within partially hyperbolic systems with one-dimensional center.

Abstract

One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile. We build a C1-open set U of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a C-infinity-dense subset of U, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center. R\'esum\'e : Dynamique mod\'er\'ee et transitivit\'e robuste. L'un des buts des syst\`emes dynamiques consiste \`a trouver une bonne d\'ecomposition de la dynamique en pi\`eces \'el\'ementaires. Cet article contribue \`a l'\'etude des classes de r\'ecurrence par cha\^ines. On sait que C1-g\'en\'eriquement, chaque classe de r\'ecurrence par cha\^ines contenant une orbite p\'eriodique coincide avec la classe homocline de cette orbite. Notre r\'esultat montre que cette propri\'et\'e est en g\'en\'erale fragile. Nous construisons un ouvert U de diff\'eomorphismes mod\'er\'es (leur dynamique ne se d\'ecompose qu'en un nombre fini de classes de r\'ecurrence par cha\^ines) tel que pour tout diff\'eomorphisme appartenant \`a un sous-ensemble C-infini-dense de U, une des classes de r\'ecurrence par cha\^ines n'est pas transitive (elle a un point isol\'e). De plus, ces dynamiques sont obtenues comme syst\`emes partiellement hyperboliques avec une direction centrale uni-dimensionnelle.