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

Abstract

Let ( X, f) be a compact topological dynamical system. Assume that in X there exist pairwise disjoint closed subsets A 1, A 2, … , A k satisfying f ( A i ) ⊇ ∪ j = 1 k A j , i = 1 , 2 , … , k . Under this assumption, the authors proved that (2 X , 2 f ), 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 (2 X , 2 f ) are actually topologically conjugate to the symbolic dynamical system ( Σ( k), σ).