Abstract
This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincaré section or completely outside a Poincaré section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give some examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.
Highlights
Ergodic theory is one of the most important observations in mathematics made in the last century with signi cance in many areas of physics such as statistical physics
We prove in the rst part of the paper that (Ω, F, P,n 0) is ergodic if and only if (Ω, F, μs, (Θnτ )n 0) is ergodic, i.e. the dynamical system (Ω, F, {μs}s∈R, (Θt)t 0) is PS-ergodic
We prove that if a periodic measure is ergodic, the generated sublinear expectation, which is invariant with respect to the skew product dynamical system or the Markov semigroup, is ergodic
Summary
Ergodic theory is one of the most important observations in mathematics made in the last century with signi cance in many areas of physics such as statistical physics (cf [4, 30,31,32]). We continue the study on the ergodicity of periodic measures and obtain some new results. We prove that if a periodic measure is ergodic, the generated sublinear expectation, which is invariant with respect to the skew product dynamical system or the Markov semigroup, is ergodic. As for the Birkhoff’s type of ergodic theorem, i.e. the law of large number, we obtain the convergence in the quasi-sure sense when we apply the ergodic theory of upper expectations, whilst we can only obtain the convergence in the almost-sure sense by the ergodic theory of periodic measures ([18]) This provides justi cations for the construction of upper expectation and the investigation of its ergodicity, which can provide useful new information. Ergodicity of skew product dynamical systems: necessary and sufficient conditions
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