In this talk, I will discuss the ideas in the development of ergodic theory under the sublinear expectation space and its analogue upper probabilities. In mind with irreducibility as its key essence, ergodicity is defined as that any invariant set has upper probability 0 or its complement upper probability 0 (Feng-Zhao (SIMA 2021/ Preprint 2017 arXiv:1705.03549)). The ergodicity is equivalent to the irreducibility of the measurable dynamical systems, the eigenvalue 1 of the Koopman operator being simple, Birkhoff's law of large numbers with single value. Under sublinear Markov setup, the theory was also developed via a corresponding lifting canonical dynamical system. It is also proved that the G-Brownian motion on the unit circle is ergodic as an example. Following this initial work, more recently a number of progresses have been obtained including Feng-Wu-Zhao (SPA 2020) on capacity, Ma-Zhao (Preprint 2024) on ergodic controls, Zhao-Zhao (Preprint 2024) on ergodicity of G-diffusions, and Feng-Huang-Liu-Zhao (Preprints 2023, 2024 arXiv:2411.00663; arXiv:2411.02030) described below. In this talk I can only concentrate a few results in the latter. First, the ergodicity is equivalent to that an invariant skeleton measure exists, is unique and ergodic. The invariant skeleton then gives the precise formula of space average as the limit of time averaging in the Birkhoff type law of large numbers. It also characterizes ergodicity with weak independence. Moreover, with the equivalent condition on ergodicity, we define weakly mixing as when the eigenvalue 1 is unique and give equivalent conditions such as ergodicity on product space and asymptotic independence. I will also discuss a weaker regime that any invariant set has upper probability 0 or 1 of Cerreia-Vioglio, Maccheroni and Marinacci (2016), in parallel to an ergodicity definition of a classical probability. We found that this does not give the irreducibility, however, is equivalent to $V$ being of finite ergodic components, Birkhoff's ergodic theorem with finite multiple values and the eigenvalue 1 of the Koopman operator being of finite multiplicity.
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