Conditions ensuring that hyperspace dynamical systems contain subsystems topologically (semi-)conjugate to symbolic dynamical systems

Let (X, f) be a compact topological dynamical system. Assume that in X there exist pairwise disjoint closed subsets A1, A2, … , Ak satisfying f(Ai)⊇∪j=1kAj,i=1,2,…,k. Under this assumption, the authors proved that (2X, 2f), which is the induced hyperspace topological dynamical system of (X, f), has at least one subsystem that is topologically semiconjugate to the symbolic dynamical system (Σ(k), σ). Further, the authors studied the properties of such subsystems and established two necessary and sufficient conditions as well as two sufficient conditions to determine which of these existing subsystems of (2X, 2f) are actually topologically conjugate to the symbolic dynamical system (Σ(k), σ).