artículo
On emergence and complexity of ergodic decompositions
Fecha
2021Registro en:
10.1016/j.aim.2021.107904
10902082
10902082 00018708
SCOPUS_ID:85112495340
WOS:000701406300005
Autor
Berger P.
Bochi J.
Institución
Resumen
© 2021 Elsevier Inc.A concept of emergence was recently introduced in [5] 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 ϵ-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 [5]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms.