Open questions leading to a global perspective in dynamics

Abstract

In this paper, I address one of the most challenging and central problems in dynamical systems, which here means flows, diffeomophisms or, more generally, transformations, defined on a closed manifold (compact, without boundary or an interval on the real line): can we describe the behaviour in the long run of typical trajectories for the ‘majority’ of systems? Poincaré was probably the first to point in this direction and stress its importance. Certainly this is so in the context of modern dynamics, whose foundation we attribute to him. Alongside Poincaré, this sense of dynamical globality can be seen in the work of Birkhoff, Morse, Andronov and Pontyagin, Peixoto, among others, and most notably Smale in the early 1960s. Smale conjectured that the limit set of all trajectories of a typical dynamical system should display a hyperbolic behaviour: along trajectories, distances increase and decrease exponentially in complementary dimensions, or in complementary dimensions transversally to flow trajectories. By the end of the decade, a number of counter examples to the conjecture were provided. Quite a few years afterwards, in 1995, see Palis (2000 Asterisque 261 339–51), based on my previous work with Takens, Newhouse, Viana and Yoccoz and several other colleagues' work, I was able to set up a program of interrelated conjectures aimed at describing the behaviour in the long run of a typical trajectory of a typical system in finite-dimensional parameter families of dynamics, the families also being typical. In brief, for a typical dynamical system, almost all trajectories have only finitely many choices, of (transitive) attractors, where to accumulate upon in the future and such attractors should be stochastically stable. Different to the more common topological viewpoint in the 1960s, the approach here is a probabilistic one, as publicized by the Russian school many years ago, and typicality is taken in terms of Lebesgue probability both in parameter and phase spaces. There is also a flavour of randomness in the behaviour at large of a typical trajectory, and of sensitivity with respect to initial conditions, unless the associated limit set attractor is just a fixed point or periodic attracting trajectory. Hence, we are also proposing a description of most chaotic systems. A strategy to verify the validity of such a global scenario is to show that this is indeed the case in the absence, in a robust way, of homoclinic tangencies, introduced by Poincaré, or heterodimensional cycles, introduced by Newhouse and myself, and much developed by Bonatti, Diaz, Rocha and others. Some of the considerable progress made along the main conjecture and related ones are commented on in this paper.