Embedding of topological dynamical systems into symbolic dynamical systems: A necessary and sufficient condition

Embedding of a topological dynamical system into another is a weaker condition than topological conjugacy between the two topological dynamical systems. In 2000, under the assumption of ε-expansive, compact and totally disconnected systems, Shigeo Akashi found a sufficient condition for embedding a topological dynamical system (X,. d, f) into a symbolic dynamical system. The purpose of this paper is to present and prove a necessary and sufficient condition that determines exactly which of the topological dynamical systems can be embedded into symbolic dynamical systems and which ones cannot be. Particularly, Shigeo Akashiʹs result becomes a special case of this new sufficient condition.