Orderings of the rationals and dynamical systems

Claudio Bonanno¹ ; Stefano Isola²

¹ Dipartimento di Matematica Applicata Università di Pisa via F. Buonarroti 1//c I-56127 Pisa, Italy
² Dipartimento di Matematica e Informatica Università di Camerino via Madonna delle Carceri I-62032 Camerino, Italy

Colloquium Mathematicum, Tome 116 (2009) no. 2, pp. 165-189.

Résumé

This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into three parts. The first one is mainly expository and consists in a critical review of rather standard topics such as Stern–Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce two classes of (invertible and non-invertible) one-dimensional maps which can be used to generate the binary trees in different ways and study their ergodic properties. This also leads us to study, in the third part, some random processes (Markov chains and martingales) which arise in a natural way from the action of the transfer operators associated to the non-invertible maps.

Détail

DOI : 10.4064/cm116-2-3

Keywords: paper devoted systematic study class binary trees encoding structure rational numbers arithmetic dynamical point view paper divided three parts first mainly expository consists critical review rather standard topics stern brocot farey trees their connections continued fraction expansion question mark function second part introduce classes invertible non invertible one dimensional maps which generate binary trees different ways study their ergodic properties leads study third part random processes markov chains martingales which arise natural action transfer operators associated non invertible maps