Introduction. Ergodic sets were introduced by Kryloff and Bogoliouboff in 1937 in connection with their study of compact dynamical systems [16]. The purpose of this paper is to review some of the work that has since been done on the theory that centers around this notion, and to present a number of supplementary remarks, applications, and simplifications. For simplicity we shall confine attenti...

The field is a relatively recently established subfield of ergodic theory, and has significant input from the two well-established areas of symbolic dynamics and Lagrangian dynamics. The large-scale picture of the field is that one is interested in optimizing potential functions over the (typically highly complex) class of invariant measures for a dynamical system. Tools that have been employed...

Every one of the important strong limit theorems that we have seen thus far – the strong law of large numbers, the martingale convergence theorem, and the ergodic theorem – has relied in a crucial way on a maximal inequality. This is no accident: it can in fact be shown that a maximal inequality is a necessary condition for an almost everywhere convergence theorem. We will refrain from carrying...