Piecewise expanding maps: combinatorics, dynamics and representation of rational numbers

Abstract

We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description of periodic and pre-periodic points, primitiveness of truncated itineraries and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of combinatorial objects as necklaces and words. We identify in a natural way each periodic orbit with an aperiodic necklace. We show the relevance of this combinatorial approach for the representation of rational numbers and for the orbit structure of pre-periodic points.

Keywords

• piecewise expanding map
• periodic point
• pre-periodic point
• representation of rational numbers