Topological conjugacy of piecewise monotonic functions of nonmonotonicity height ≥1

The conjugacy problem is an important topic in the theory of dynamical systems and functional equations. In this paper, we investigate a class of piecewise monotone and continuous maps with nonmonotonicity height ≥1. We give a sufficient and necessary condition under which any two of these maps are topologically conjugate, and construct a topological conjugacy with an extension method if such a conjugacy exists.