  A concept of emergence was recently introduced in the paper [Berger] in order
to quantify the richness of possible statistical behaviors of orbits of a given
dynamical system. In this paper, we develop this concept and provide several
new definitions, results, and examples. We introduce the notion of topological
emergence of a dynamical system, which essentially evaluates how big the set of
all its ergodic probability measures is. On the other hand, the metric
emergence of a particular reference measure (usually Lebesgue) quantifies how
non-ergodic this measure is. We prove fundamental properties of these two
emergences, relating them with classical concepts such as Kolmogorov's
$\epsilon$-entropy of metric spaces and quantization of measures. We also
relate the two types of emergences by means of a variational principle.
Furthermore, we provide several examples of dynamics with high emergence.
First, we show that the topological emergence of some standard classes of
hyperbolic dynamical systems is essentially the maximal one allowed by the
ambient. Secondly, we construct examples of smooth area-preserving
diffeomorphisms that are extremely non-ergodic in the sense that the metric
emergence of the Lebesgue measure is essentially maximal. These examples
confirm that super-polynomial emergence indeed exists, as conjectured in the
paper [Berger]. Finally, we prove that such examples are locally generic among
smooth diffeomorphisms.
