TY - JOUR

T1 - On emergence and complexity of ergodic decompositions

AU - Berger, Pierre

AU - Bochi, Jairo

N1 - Funding Information:
Partially supported by the ERC project 818737 Emergence of wild differentiable dynamical systems.Partially supported by project FONDECYT 1180371 and project CONICYT PIA ACT172001 New trends in ergodic theory.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/10/29

Y1 - 2021/10/29

N2 - 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.

AB - 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.

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U2 - 10.1016/j.aim.2021.107904

DO - 10.1016/j.aim.2021.107904

M3 - Article

AN - SCOPUS:85112495340

VL - 390

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107904

ER -