[Paper Review] Tame dynamics and robust transitivity

Christian Bonatti, Sylvain Crovisier+2 more|arXiv (Cornell University)|Dec 5, 2011

Mathematical Dynamics and Fractals19 references3 citations

TL;DR

This paper demonstrates that robust transitivity in chain-recurrence classes is not generic among tame diffeomorphisms, even in the $C^1$-topology. Using a construction of partially hyperbolic diffeomorphisms with one-dimensional center, the authors exhibit a $C^1$-open set of tame systems where robustly isolated chain-recurrence classes fail to be transitive under $C^r$-perturbations for any $r > 1$, showing that the property of robust transitivity is fragile in this setting.

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.

Motivation & Objective

• To investigate whether robustly isolated chain-recurrence classes in tame diffeomorphisms are necessarily robustly transitive.
• To determine whether the property that chain-recurrence classes coincide with homoclinic classes is stable under small $C^r$-perturbations.
• To construct explicit examples of $C^1$-open sets of tame diffeomorphisms where robustly isolated classes are not transitive.
• To analyze the dynamics of partially hyperbolic systems with one-dimensional center in relation to chain-recurrence and transitivity.

Proposed method

• Construct a $C^1$-open set $\mathcal{U}$ of diffeomorphisms on a compact manifold $M$ with $\dim(M) \geq 3$, where the dynamics are tame and partially hyperbolic with one-dimensional center.
• Define a robustly isolated chain-recurrence class $\mathcal{C}_f$ for a diffeomorphism $f \in \mathcal{U}$, ensuring the class persists under $C^1$-perturbations.
• Use local dynamics near hyperbolic periodic orbits $p$ and $q$, with stable indices 2 and 1 respectively, to analyze intersections of invariant manifolds.
• Apply the $\lambda$-lemma and transversality arguments to show that $W^{s}(p)$ and $W^{u}(q)$ intersect transversally under perturbations.
• Prove that the set of diffeomorphisms for which $\mathcal{C}_g$ is not transitive forms a $C^r$-dense subset in a $C^1$-neighborhood of $f$, using codimension-one submanifold arguments.
• Characterize the non-transitive part of $\mathcal{C}_f$ as $W^s(q) \cap W^u(p)$, showing it consists of isolated points not in the homoclinic class of $p$ or $q$.

Experimental results

Research questions

• RQ1 Is every robustly isolated chain-recurrence class in a tame diffeomorphism necessarily robustly transitive?
• RQ2 Can a $C^1$-open set of tame diffeomorphisms contain a robustly isolated chain-recurrence class that fails to be transitive under $C^r$-perturbations for $r > 1$?
• RQ3 To what extent is the coincidence of chain-recurrence classes with homoclinic classes stable under small $C^r$-perturbations?
• RQ4 How do the dynamics of partially hyperbolic systems with one-dimensional center affect the transitivity of chain-recurrence classes?
• RQ5 What is the structure of the non-transitive part of a robustly isolated chain-recurrence class in such systems?

Key findings

• There exists a $C^1$-open set $\mathcal{U}$ of tame diffeomorphisms on any compact manifold $M$ with $\dim(M) \geq 3$ such that for any $f \in \mathcal{U}$, the chain-recurrence class $\mathcal{C}_f$ is robustly isolated.
• For any $r > 1$, the set of diffeomorphisms $g$ in a $C^1$-neighborhood of $f$ for which $\mathcal{C}_g$ is not transitive is $C^r$-dense.
• The non-transitive part of $\mathcal{C}_f$ is exactly $W^s(q) \cap W^u(p)$, which consists of isolated points and is disjoint from the homoclinic class of $p$.
• The homoclinic class of $p$ coincides with the homoclinic class of $q$, and contains all periodic points in $\mathcal{C}_f$ of the same stable index.
• Any point in $\mathcal{C}_f \setminus H_f$ belongs to the homoclinic class of $p$ if it is not in $W^s(q)$, and to the homoclinic class of $q$ if not in $W^u(p)$.
• The set of diffeomorphisms where $\mathcal{C}_g$ fails to be transitive is a countable union of codimension-one submanifolds in the $C^1$-topology, implying $C^r$-density for $r > 1$.