##
**A global view of dynamics and a conjecture on the denseness of finitude of attractors.**
*(English)*
Zbl 1044.37014

Flexor, Marguerite (ed.) et al., Complex geometry and dynamical systems. Conference in honor of Adrien Douady on the occasion of his 60th birthday, Orsay, France, July 3–8, 1995. Paris: Astérisque. Astérisque 261, 335-347 (2000).

Summary: A view on dissipative dynamics, i.e. flows, diffeomorphisms, and transformations in general of a compact boundaryless manifold or the interval is presented here, including several recent results, open problems and conjectures. It culminates with a conjecture on the denseness of systems having only finitely many attractors, the attractors being sensitive to initial conditions (chaotic) or just periodic sinks and the union of their basins of attraction having total probability. Moreover, the attractors should be stochastically stable in their basins of attraction. This formulation, dating from early 1995, sets the scenario for the understanding of most nearby systems in parametrized form. It can be considered as a probabilistic version of the once considered possible existence of an open and dense subset of systems with dynamically stable structures, a dream of the sixties that evaporated by the end of that decade. The collapse of such a previous conjecture excluded the case of one dimensional dynamics: it is true at least for real quadratic maps of the interval as shown independently by Swiatek, with the help of Graczyk [J. Graczyk and G. Swiatek, Ann. Math. (2) 146, 1–52 (1997; Zbl 0936.37015)], and M. Lyubich [Acta Math. 178, 185–297 (1997; Zbl 0908.58053)] a few years ago. Recently, O. Kozlovski [Structural stability in one-dimensional dynamics, PhD thesis, Univ. Amsterdam (1997)] announced the same result for \(C^3\) unimodal mappings, in a meeting at IMPA. Actually, for one-dimensional real or complex dynamics, our main conjecture goes even further: for most values of parameters, the corresponding dynamical system displays finitely many attractors which are periodic sinks or carry an absolutely continuous invariant probability measure. Remarkably, M. Lyubich [Ann. Math. (2) 156, No. 1, 1–78 (2002; Zbl 1160.37356)] has just proved this for the family of real quadratic maps of the interval, with the help of M. Martens and T. Nowicki [Astérisque 261, 239–252 (2000; Zbl 0939.37020)].

For the entire collection see [Zbl 0932.00046].

For the entire collection see [Zbl 0932.00046].

### MSC:

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37C20 | Generic properties, structural stability of dynamical systems |

37C75 | Stability theory for smooth dynamical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |

37G25 | Bifurcations connected with nontransversal intersection in dynamical systems |