Universality with respect to ω-limit sets

A discrete dynamical system on a compact metric space X is called universal (with respect to ω-limit sets) if, among its ω-limit sets, there is a homeomorphic copy of any ω-limit set of any dynamical system on X. By a result of Pokluda and Smítal the unit interval admits a universal system. In this paper, we study the problem of the existence of universal systems on Cantor spaces, graphs, dendrites and higher-dimensional spaces.