### 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 I_{α} 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 language | English |
---|---|

Pages (from-to) | 24-41 |

Number of pages | 18 |

Journal | Journal of Approximation Theory |

Volume | 162 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2010 Jan |

### Fingerprint

### Keywords

- Approximations of real numbers
- Brjuno function
- Continued fractions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Mathematics(all)
- Numerical Analysis

### Cite this

*Journal of Approximation Theory*,

*162*(1), 24-41. https://doi.org/10.1016/j.jat.2009.02.004

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

Research output: Contribution to journal › Article

*Journal of Approximation Theory*, vol. 162, no. 1, pp. 24-41. https://doi.org/10.1016/j.jat.2009.02.004

}

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 -