Attractors with Large Invisible Parts

Andrei Negut, Princeton University
Fine Hall 401

Philosophically, an attractor of a dynamical system is a closed subset of the phase space which orbits "approach" as time goes to infinity. Different meanings of the word "approach" produce different versions of attractors: maximal, Milnor, statistical, minimal etc. The question of generic non-coincidence between these various types of attractors has not yet been answered. We will be concerned with a different point of view. Physically, if one looks at the attractor, then one knows where most orbits will go to as time goes to infinity. But it is possible that a large part of the attractor is redundant, in the sense that orbits spend very very little time near it. Thus, it would be more significant to look only at the non-redundant part of the attractor. Concretely, we will present an example of a random dynamical system, given by parameters of "reasonable magnitude" (e.g. 1000). For this dynamical system, roughly half of the attractor is "invisible" in the sense that orbits spend near it only a fraction of $2^{-500}$ of all time. The number $2^{-500}$ is equal to zero for all physical or computer experiments, and therefore an observer should not bother with the "invisible" half of the attractor. Moreover, small perturbations of this dynamical system exhibit the same property, and therefore the phenomenon is generic.