Issue |
ESAIM: Proc.
Volume 46, November 2014
ECIT 2012, 19th European Conference on Iteration Theory
|
|
---|---|---|
Page(s) | 213 - 216 | |
DOI | https://doi.org/10.1051/proc/201446017 | |
Published online | 05 December 2014 |
Piecewise expanding maps: combinatorics, dynamics and representation of rational numbers
1 Centro de Matemática e Aplicações
Fundamentais, Departamento de Matemática, Faculdade de Ciências, Universidade de
Lisboa, 1749-016
Lisboa,
Portugal
Tel: 00351 2175100293, E-mail: mcserpa@fc.ul.pt
2 Centro de Matemática e Aplicações
Fundamentais, Departamento de Matemática, Faculdade de Ciências, Universidade de
Lisboa, 1749-016
Lisboa,
Portugal
Tel: 00351 2175100293, E-mail: jsbuescu@fc.ul.pt
We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description of periodic and pre-periodic points, primitiveness of truncated itineraries and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of combinatorial objects as necklaces and words. We identify in a natural way each periodic orbit with an aperiodic necklace. We show the relevance of this combinatorial approach for the representation of rational numbers and for the orbit structure of pre-periodic points.
Résumé
Nous établissons des propriétés combinatoires de la dynamique des fonctions de l’intervalle croissant, continu et expansif par morceaux tels que description des points périodiques et pré-périodiques, primitif itinéraires de troncs et de la durée d’itinéraires pré-périodiques. Nous inclure une relation entre la dynamique d’une famille de fonctions du cercle et les propriétés des objets combinatoires que des colliers et des mots. Nous identifions de façon naturelle chaque orbite périodique avec un apériodique collier. Nous montrons la pertinence de cette approche combinatoire pour la représentation des nombres rationnels et de la structure de l’orbite des points de pré-périodiques.
Mathematics Subject Classification: 37E15 / 37E05 / 37E10
Key words: Piecewise expanding map / periodic point / pre-periodic point / representation of rational numbers
Mots clés : Fonction expansif par morceaux / point périodique / point pré-périodique / représentation des nombres rationnels
© EDP Sciences, SMAI 2014
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