Ergodicityof N-continuedfractionexpansions

K. Dajani; C. Kraaikamp; N. van der Wekken

doi:10.1016/j.jnt.2013.02.017

Ergodicityof N-continuedfractionexpansions

K. Dajani^*, C. Kraaikamp, N. van der Wekken

^*Corresponding author for this work

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Research output: Contribution to journal › Article › Academic › peer-review

Abstract

Recently,EdwardBurgerandhisco-authorsintroducedandstudied in Burgeretal.(2008) [3] a newclassofcontinuedfraction algorithms.Inparticulartheyshowedthatforeveryquadratic irrationalnumber x thereexistinfinitelymanyeventuallyperiodic N-expansionswithperiod-length1;seealsoKomatsu(2009) [10] forrelatedproperties.In2011,MaxwellAnselmandSteven Weintraubfurtherstudiedthepropertiesof N-expansionsin Anselm andWeintraub(2011) [2]. Oneniceresulttheyobtainedis that every x between0and N has uncountablymany N-expansions for eachinteger N 2. Inthispaperwewillreprovethisresultand fromthiswestudytheergodicpropertiesofvarioussubclassesof N-expansions.

Original language	English
Pages (from-to)	3183-3204
Number of pages	20
Journal	Journal of Number Theory
Volume	133
Issue number	9
DOIs	https://doi.org/10.1016/j.jnt.2013.02.017
Publication status	Published - Sept 2013

Keywords

Continued fractions
Ergodicity
σ-Finite
infinite invariant measure

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@article{4b97e1e9130a44c4b2b78f5d4f89b26a,

title = "Ergodicityof N-continuedfractionexpansions",

abstract = "Recently,EdwardBurgerandhisco-authorsintroducedandstudied in Burgeretal.(2008) [3] a newclassofcontinuedfraction algorithms.Inparticulartheyshowedthatforeveryquadratic irrationalnumber x thereexistinfinitelymanyeventuallyperiodic N-expansionswithperiod-length1;seealsoKomatsu(2009) [10] forrelatedproperties.In2011,MaxwellAnselmandSteven Weintraubfurtherstudiedthepropertiesof N-expansionsin Anselm andWeintraub(2011) [2]. Oneniceresulttheyobtainedis that every x between0and N has uncountablymany N-expansions for eachinteger N 2. Inthispaperwewillreprovethisresultand fromthiswestudytheergodicpropertiesofvarioussubclassesof N-expansions.",

keywords = "Continued fractions, Ergodicity, σ-Finite, infinite invariant measure",

author = "K. Dajani and C. Kraaikamp and {van der Wekken}, N.",

year = "2013",

month = sep,

doi = "10.1016/j.jnt.2013.02.017",

language = "English",

volume = "133",

pages = "3183--3204",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "9",

}

TY - JOUR

T1 - Ergodicityof N-continuedfractionexpansions

AU - Dajani, K.

AU - Kraaikamp, C.

AU - van der Wekken, N.

PY - 2013/9

Y1 - 2013/9

N2 - Recently,EdwardBurgerandhisco-authorsintroducedandstudied in Burgeretal.(2008) [3] a newclassofcontinuedfraction algorithms.Inparticulartheyshowedthatforeveryquadratic irrationalnumber x thereexistinfinitelymanyeventuallyperiodic N-expansionswithperiod-length1;seealsoKomatsu(2009) [10] forrelatedproperties.In2011,MaxwellAnselmandSteven Weintraubfurtherstudiedthepropertiesof N-expansionsin Anselm andWeintraub(2011) [2]. Oneniceresulttheyobtainedis that every x between0and N has uncountablymany N-expansions for eachinteger N 2. Inthispaperwewillreprovethisresultand fromthiswestudytheergodicpropertiesofvarioussubclassesof N-expansions.

AB - Recently,EdwardBurgerandhisco-authorsintroducedandstudied in Burgeretal.(2008) [3] a newclassofcontinuedfraction algorithms.Inparticulartheyshowedthatforeveryquadratic irrationalnumber x thereexistinfinitelymanyeventuallyperiodic N-expansionswithperiod-length1;seealsoKomatsu(2009) [10] forrelatedproperties.In2011,MaxwellAnselmandSteven Weintraubfurtherstudiedthepropertiesof N-expansionsin Anselm andWeintraub(2011) [2]. Oneniceresulttheyobtainedis that every x between0and N has uncountablymany N-expansions for eachinteger N 2. Inthispaperwewillreprovethisresultand fromthiswestudytheergodicpropertiesofvarioussubclassesof N-expansions.

KW - Continued fractions

KW - Ergodicity

KW - σ-Finite

KW - infinite invariant measure

U2 - 10.1016/j.jnt.2013.02.017

DO - 10.1016/j.jnt.2013.02.017

M3 - Article

SN - 0022-314X

VL - 133

SP - 3183

EP - 3204

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 9

ER -

Ergodicityof N-continuedfractionexpansions

Abstract

Keywords

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