Generalized Reversibility of Topological Dynamical Systems and Cellular Automata

In this paper, we characterize the reversibility of topological dynamical systems over the limit sets. We define a new concept of generalized inverse systems for topological dynamical systems, and prove that (i) a topological dynamical system has a generalized inverse system if and only if it is injective over its limit set and its limit set is reached in finite time, and (ii) if a topological dynamical system has a generalized inverse system, these two systems have the same topological entropy. For cellular automata (CAs), a particular class of topological dynamical systems, we prove some additional properties: (iii) A CA has a generalized inverse CA if and only if it is injective over its limit set. (iv) It is undecidable whether a given CA has a generalized inverse CA.