Piecewise monotone maps without periodic points: rigidity, measures and complexity

Abstract

We consider piecewise monotone maps of the interval with zero entropy or no periodic points. First, we give a rigid model for these maps: the interval translations mappings, possibly with flips. It follows, for example, that the complexity of a piecewise monotone map of the interval is at most polynomial if and only if this map has a finite number of periodic points up to monotone equivalence. Second, we study the invariant and ergodic measures of a piecewise monotone map with zero entropy and prove that their number is bounded by twice the number of monotony intervals; for a piecewise increasing map their number is at most the number of intervals.