Generalized Brjuno functions associated to α-continued fractions

Laura Luzzi, Stefano Marmi, Hitoshi Nakada, Rie Natsui

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), the nearest integer (α = 1 / 2) and by-excess (α = 0) continued fraction maps. To each of these expansions and to each choice of a positive function u on the interval Iα we associate a generalized Brjuno function B(α, u) (x). When α = 1 / 2 or α = 1, and u (x) = - log (x), these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of α and we prove that the set of (α, u)-Brjuno numbers does not depend on the choice of α provided that α ≠ 0. We then consider the case α = 0, u (x) = - log (x) and we prove that x is a Brjuno number (for α ≠ 0) if and only if both x and - x are Brjuno numbers for α = 0.

Original languageEnglish
Pages (from-to)24-41
Number of pages18
JournalJournal of Approximation Theory
Volume162
Issue number1
DOIs
Publication statusPublished - 2010 Jan

Fingerprint

Continued fraction
Generalized Functions
Gauss Map
Holomorphic Maps
Linearization
Excess
If and only if
Interval
Integer

Keywords

  • Approximations of real numbers
  • Brjuno function
  • Continued fractions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)
  • Numerical Analysis

Cite this

Generalized Brjuno functions associated to α-continued fractions. / Luzzi, Laura; Marmi, Stefano; Nakada, Hitoshi; Natsui, Rie.

In: Journal of Approximation Theory, Vol. 162, No. 1, 01.2010, p. 24-41.

Research output: Contribution to journalArticle

Luzzi, Laura ; Marmi, Stefano ; Nakada, Hitoshi ; Natsui, Rie. / Generalized Brjuno functions associated to α-continued fractions. In: Journal of Approximation Theory. 2010 ; Vol. 162, No. 1. pp. 24-41.
@article{88fee27a577d4aa4ab5fa5621c6555c5,
title = "Generalized Brjuno functions associated to α-continued fractions",
abstract = "For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), the nearest integer (α = 1 / 2) and by-excess (α = 0) continued fraction maps. To each of these expansions and to each choice of a positive function u on the interval Iα we associate a generalized Brjuno function B(α, u) (x). When α = 1 / 2 or α = 1, and u (x) = - log (x), these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of α and we prove that the set of (α, u)-Brjuno numbers does not depend on the choice of α provided that α ≠ 0. We then consider the case α = 0, u (x) = - log (x) and we prove that x is a Brjuno number (for α ≠ 0) if and only if both x and - x are Brjuno numbers for α = 0.",
keywords = "Approximations of real numbers, Brjuno function, Continued fractions",
author = "Laura Luzzi and Stefano Marmi and Hitoshi Nakada and Rie Natsui",
year = "2010",
month = "1",
doi = "10.1016/j.jat.2009.02.004",
language = "English",
volume = "162",
pages = "24--41",
journal = "Journal of Approximation Theory",
issn = "0021-9045",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - Generalized Brjuno functions associated to α-continued fractions

AU - Luzzi, Laura

AU - Marmi, Stefano

AU - Nakada, Hitoshi

AU - Natsui, Rie

PY - 2010/1

Y1 - 2010/1

N2 - For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), the nearest integer (α = 1 / 2) and by-excess (α = 0) continued fraction maps. To each of these expansions and to each choice of a positive function u on the interval Iα we associate a generalized Brjuno function B(α, u) (x). When α = 1 / 2 or α = 1, and u (x) = - log (x), these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of α and we prove that the set of (α, u)-Brjuno numbers does not depend on the choice of α provided that α ≠ 0. We then consider the case α = 0, u (x) = - log (x) and we prove that x is a Brjuno number (for α ≠ 0) if and only if both x and - x are Brjuno numbers for α = 0.

AB - For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), the nearest integer (α = 1 / 2) and by-excess (α = 0) continued fraction maps. To each of these expansions and to each choice of a positive function u on the interval Iα we associate a generalized Brjuno function B(α, u) (x). When α = 1 / 2 or α = 1, and u (x) = - log (x), these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of α and we prove that the set of (α, u)-Brjuno numbers does not depend on the choice of α provided that α ≠ 0. We then consider the case α = 0, u (x) = - log (x) and we prove that x is a Brjuno number (for α ≠ 0) if and only if both x and - x are Brjuno numbers for α = 0.

KW - Approximations of real numbers

KW - Brjuno function

KW - Continued fractions

UR - http://www.scopus.com/inward/record.url?scp=70450239875&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70450239875&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2009.02.004

DO - 10.1016/j.jat.2009.02.004

M3 - Article

VL - 162

SP - 24

EP - 41

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 1

ER -