Orderings of the rationals and dynamical systems

• Autorzy: Claudio Bonanno , Stefano Isola
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37A30: Ergodic theorems, spectral theory, Markov operators
• 11B57: Farey sequences; the sequences 1 k , 2 k , ⋯
• 11A55: Continued fractions
• 37E25: Maps of trees and graphs
• 37A45: Relations with number theory and harmonic analysis
• 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2009
• Tom: 116
• Numer: 2
• Strony: 165-189

Daty

wydano

2009

Twórcy

autor

Claudio Bonanno

• Dipartimento di Matematica Applicata, Università di Pisa, via F. Buonarroti 1/c, I-56127 Pisa, Italy

autor

Stefano Isola

• Dipartimento di Matematica e Informatica, Università di Camerino, via Madonna delle Carceri, I-62032 Camerino, Italy

Identyfikatory

DOI

10.4064/cm116-2-3