Topological conjugacy of PM functions with height equaling ∞

It is known that topological conjugacy is a basic equivalence relation in dynamical systems. In this paper we study a class of piecewise monotone and continuous functions with infinite height. Those functions are topologically conjugate with each other if and only if they have same sequences describing itineraries of all forts, endpoints, and fixed points. We construct the topological conjugacy by extension, which partly generalizes previous results.