Robust transitivity in hamiltonian dynamics

Nassiri, Meysam; Pujals, Enrique R.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 2, pp. 191-239.

Un objectif de ce travail est d’étudier la dynamique sur le complémentaire des tores KAM en mettant l’accent sur la transitivité robuste non locale. Nous introduisons les ensembles ouverts de difféomorphismes symplectiques et de systèmes hamiltoniens, présentant de grands ensembles robustement transitifs. L’adhérence de ces ensembles ouverts (en topologie $C^r$, $r=1,2,\cdots ,\infty$) contient un grand nombre de systèmes, y compris les systèmes intégrables a priori instables. En outre, l'existence de mesures ergodiques avec un grand support est obtenue pour l'ensemble de ces systèmes. L'ingrédient principal des preuves est la combinaison de l'étude de systèmes itérés de fonctions de dynamique minimale et d'un nouvel outil de la dynamique hamiltonienne que nous appelons « mélangeurs symplectiques ».

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1,2,\cdots ,\infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting large robustly transitive sets. We show that the $C^{\infty }$ closure of such open sets contains a variety of systems, including so-called a priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the proof is a combination of studying minimal dynamics of symplectic iterated function systems and a new tool in Hamiltonian dynamics which we call “symplectic blender”.

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