Mean Li-Yorke chaos of random dynamical systems for amenable groups

In this paper, we investigate mean Li-Yorke chaos of all finite orders along admissible sequences for continuous bundle random dynamical systems associated with bi-orderable amenable group actions. Let G be a countable discrete infinite bi-orderable amenable group equipped with an algebraic past Φ satisfying gΦg−1⊆Φ for all g∈G. Let F be a continuous bundle random dynamical system with Polish fibre space X over an ergodic Polish G-system (Ω,F,P,G). We prove that if E is an F-invariant random compact set satisfying htop(r)(F,E)>0, then, for every admissible sequence s=(sk)k≥1 of pairwise distinct elements of G such that (GN(s))N≥1 is a tempered Følner sequence, the following holds for P-a.s. ω∈Ω: the fibre Eω contains a Mycielski set E(ω) such that, for every integer n≥2, there exists a constant εn,ω>0 such that every n distinct points x1,…,xn∈E(ω) satisfy lim infN→∞1|GN(s)|∑g∈GN(s)max1≤i<j≤nd(Fg,ω(xi),Fg,ω(xj))=0,lim supN→∞1|GN(s)|∑g∈GN(s)min1≤i<j≤nd(Fg,ω(xi),Fg,ω(xj))>εn,ω, where d is the bounded compatible metric fixed below.