Ergodicity and Mixing of invariant capacities and applications

TL;DR

This paper extends ergodic theory to invariant capacities, introducing new notions like common conditional expectation, and applies these to establish ergodic theorems, laws of large numbers, and characterizations of mixing in non-additive probability frameworks.

Contribution

It introduces the concept of common conditional expectation for invariant capacities and develops ergodic theorems and mixing characterizations in this non-additive setting.

Findings

Established a strong law of large numbers for ergodic stationary sequences on upper probability spaces.

Proved the multiplicative ergodic theorem within the context of upper probabilities.

Provided criteria for ergodicity of upper probabilities based on independence.

Abstract

We introduce the notion of common conditional expectation to investigate Birkhoff's ergodic theorem and subadditive ergodic theorem for invariant upper probabilities. If in addition, the upper probability is ergodic, we construct an invariant probability to characterize the limit of the ergodic mean. Moreover, this skeleton probability is the unique ergodic probability in the core of the upper probability, that is equal to all probabilities in the core on all invariant sets. We have the following applications of these two theorems: $\bullet$ provide a strong law of large numbers for ergodic stationary sequence on upper probability spaces; $\bullet$ prove the multiplicative ergodic theorem on upper probability spaces; $\bullet$ establish a criterion for the ergodicity of upper probabilities in terms of independence. Furthermore, we introduce and study weak mixing for capacity preserving systems. Using the skeleton idea, we also provide several characterizations of weak mixing for invariant upper probabilities. Finally, we provide examples of ergodic and weakly mixing capacity preserving systems. As applications, we obtain new results in the classical ergodic theory. e.g. in characterizing dynamical properties on measure preserving systems, such as weak mixing, periodicity. Moreover, we use our results in the nonlinear theory to obtain the asymptotic independence, Birkhoff's type ergodic theorem, subadditive ergodic theorem, and multiplicative ergodic theorem for non-invariant probabilities.