Orderings of the rationals and dynamical systems

Claudio 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 -