Mean Topological Dimension And Relative Tail Entropy For Random Bundle Transformations

We discuss two topics in this paper.One is about mean dimension and the other is about tail entropy.Both of them are established on continuous bundle random dynamical systems（RDS）.We introduce the notion of mean topological dimension for continuous bundle RDS which ismdim（ε,T）sup α∈D∫D（α,T,ω）dP,where D is a special class of countable random open covers of ε and D（α,T,ω）is the P-allost all pointwise limit of 1/n D（αn（ω））.We define the concept of orbit capacity,small set and the small boundary property for RDS.We prove that ∫ ocap（E,ω）dP = sup {μE):μ∈ LP（ε）}for random closed set E.We also show the relationship between mean dimension and the small boundary property.We obtain a variational inequality hm*（Γ|DH）≤h*（T）for product random dynam-ics,where Γ is the skew product map of the product system and DH is the pull-back a-algebra of the natural measurable stucture on the first bundle.We also construct a maximal invariant measure to ensure that the relative tail entropy could be reached when the two continuous bundle RDSs coincide.max {hμ*(（?）^（2）|A_ε（2）:μ∈LP（ε）}=h*（T）.