# Orderings of the rationals and dynamical systems

Claudio Bonanno; Stefano Isola

Colloquium Mathematicae (2009)

- Volume: 116, Issue: 2, page 165-189
- ISSN: 0010-1354

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topClaudio Bonanno, and Stefano Isola. "Orderings of the rationals and dynamical systems." Colloquium Mathematicae 116.2 (2009): 165-189. <http://eudml.org/doc/283572>.

@article{ClaudioBonanno2009,

abstract = {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.},

author = {Claudio Bonanno, Stefano Isola},

journal = {Colloquium Mathematicae},

keywords = {Stern–Brocot tree; continued fractions; question mark function; rank-one transformations; transfer operators; martingales},

language = {eng},

number = {2},

pages = {165-189},

title = {Orderings of the rationals and dynamical systems},

url = {http://eudml.org/doc/283572},

volume = {116},

year = {2009},

}

TY - JOUR

AU - Claudio Bonanno

AU - Stefano Isola

TI - Orderings of the rationals and dynamical systems

JO - Colloquium Mathematicae

PY - 2009

VL - 116

IS - 2

SP - 165

EP - 189

AB - 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.

LA - eng

KW - Stern–Brocot tree; continued fractions; question mark function; rank-one transformations; transfer operators; martingales

UR - http://eudml.org/doc/283572

ER -

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