- 99-235 Jacob Palis
- A global view of Dynamics and a conjecture on the
denseness of the finitude of attractors
(89K, LaTeX2e with two SMF style files)
Jun 17, 99
(auto. generated ps),
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Abstract. 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 [GS], and Lyubich [Ly1] a few years ago.
Recently, Kozlovski [Ko] 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, Lyubich [Ly2] has just proved this for the family of
real quadratic maps of the interval, with the help of Martens and
This work was accepted for publication in Ast\'erisque about two
years ago. It is expected to appear by the end of the present year.