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 On piecewise monotone functions with height being infinity Hong Zhu,Lin Li,YingYing Zeng,Zhiheng Yu Keywords:Piecewise monotone function, height, topological conjugacy, fort. Abstract: It is known that every piecewise monotone function with height finity has a characteristic interval after finite times iteration, and then the study of dynamics for such functions is able to be restricted to their characteristic intervals, which becomes monotone case. To the opposite, the description for piecewise monotone functions with height being infinity is much more complicated since the theory of characteristic interval does not work anymore. In this paper, we consider the problem of topological conjugacy for piecewise monotone functions with height being infinity. Some necessary and sufficient conditions are given for the existence of conjugacies between these functions. Moreover, the height of infinity under composition is also discussed. The fact shows a kind of symmetry for the height.