A global view of dynamics and a conjecture on the denseness of ﬁnitude of attractors

By J. Palis. Astérisque, 2018

161 citations recorded by the source corpus.

. — A view on dissipative dynamics, i.e. flows, diffeomorphisms, and trans formations 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 attrac tors should be stochastically stable in their basins of attraction. This formulation, dating from early 1995, sets the scenario for the understanding of most nearby sys tems 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 dynam ically 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 dimen sional 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 [Lyl] a few years ago. Recently, Kozlovski [Ko] announced the same result for C3 unimodal map pings, 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 correspond ing 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 Nowicki [MN].

Subfields: Computer Networks and Communications, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ab78b03ccbdfaf7e7bbfd0f130e4bed4ec572865

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Citations in the corpus

Abundance of C 1 -robust homoclinic tangencies (2009, background): “Moreover, homoclinic bifurcations (homoclinic tangencies and heterodimensional cycles) are conjectured to be the main source of non-hyperbolic dynamics (Palis denseness conjecture, see [23]).”