Mean Li-Yorke chaos along any infinite sequence for infinite-dimensional random dynamical systems

In this paper, we study the mean Li-Yorke chaotic phenomenon along any infinite positive integer sequence for infinite-dimensional random dynamical systems. To be precise, we prove that if an injective continuous infinite-dimensional random dynamical system (X, phi) over an invertible ergodic Polish system (Omega, F, P, theta) admits a phi-invariant random compact set K with positive topological entropy, then given a positive integer sequence a = {a(i)}(i is an element of N) with lim(i ->+infinity)a(i) = +infinity, for P-a.s. omega is an element of Omega there exists an uncountable subset S(omega) subset of K(omega) and epsilon(omega) > 0 such that for any two distinct points x(1), x(2) is an element of S(omega), one has lim inf (N ->+infinity) 1/N Sigma(N)(i =1) d (phi(a(i),omega)x (1),phi(a(i),omega)x(2)) = 0, lim sup(N ->+infinity )1/N Sigma(N)(i =1)d (phi(a(i), omega)x(1), phi (a(i), omega)x(2)) > epsilon(omega), where d is a compatible complete metric on X . To overcome the difficulties arising in the infinitedimensional random dynamical systems, we use the regularity of measure to construct the expected measurable partitions and to decompose the distance function. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.