Abstract
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.