Topological and measurable universality in linear dynamics

Overview & Synopsis

This is for the most part a survey paper, in which we present some results related to the construction of universal linear dynamical systems, i.e. of operators on Banach spaces which can represent, in various senses, certain classes of (nonlinear) dynamical systems. We first describe and generalize a result of Feldman proving the existence of universal linear systems in the topological sense. We then present the first example, due to Glasner and Weiss, of a universal linear system in the measurable framework. After this we report on a recent generalization of the Glasner-Weiss universality result, giving a simple and general criterion for an operator on a separable Banach space to be universal. The main ideas of the proof of this criterion are also presented.