Semiconjugacy to a map of a constant slope

Abstract

It is well known that a continuous piecewise monotoneinterval map with positive topological entropy is semiconjugate to amap of a constant slope and the same entropy, and if it isadditionally transitive then this semiconjugacy is actually aconjugacy. We generalize this result topiecewise continuous piecewise monotone interval maps, andas a consequence, get it also for piecewise monotone graph maps.We show that assigning to a continuous transitive piecewisemonotone map of positive entropy a map of constant slope conjugate toit defines an operator, and show that this operator is not continuous.