A complete classification of the piecewise monotone functions on the interval

Abstract

We define two functions f f and g g on the unit interval [ 0 , 1 ] [0,1] to be strongly conjugate iff there is an order-preserving homeomorphism h h of [ 0 , 1 ] [0,1] such that g = h − 1 f h g = {h^{ - 1}}fh (a minor variation of the more common term "conjugate", in which h h need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.