Dynamical directions in numeration

Abstract

This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -numeration and its generalisations, abstract numeration systems and shift radix systems, as well as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> -scales and odometers. A section of applications ends the paper.