A Complete Classification of the Piecewise Monotone Functions on the Interval

Abstract

We define two functions $f$ and $g$ on the unit interval $[0,1]$ to be strongly conjugate iff there is an order-preserving homeomorphism $h$ of $[0,1]$ such that $g = {h^{ - 1}}fh$ (a minor variation of the more common term "conjugate", in which $h$ 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.