Metric Properties of Denjoy's Canonical Continued Fraction Expansion

Metric Properties of Denjoy's Canonical Continued Fraction Expansion

Marius IOSIFESCU and Cor KRAAIKAMP

Source: Tokyo J. of Math. Volume 31, Number 2 (2008), 495-510.

Abstract

Metric properties of Denjoy's canonical continued fraction expansion are studied, and the natural extension of the underlying ergodic system is given. This natural extension is used to give simple proofs of results on mediant convergents obtained by W. Bosma in 1990.

Primary Subjects: 28D05

Secondary Subjects: 11K55

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.tjm/1233844066
Digital Object Identifier: doi:10.3836/tjm/1233844066
Mathematical Reviews number (MathSciNet): MR2477886
Zentralblatt MATH identifier: 05545426

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References

Aaronson, J., An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997.

Mathematical Reviews (MathSciNet): MR1450400

Zentralblatt MATH: 0882.28013

Bosma, W., Approximation by mediants, Math. Comp., 54 (1990), no. 189, 421--434.

Mathematical Reviews (MathSciNet): MR995207

Zentralblatt MATH: 0697.10042

Digital Object Identifier: doi:10.2307/2008703

JSTOR: links.jstor.org

Bosma, W., Jager, H. and Wiedijk, F., Some metrical observations on the approximation by continued fractions, Nederl. Akad. Wetensch. Indag. Math., 45 (1983), no. 3, 281--299.

Mathematical Reviews (MathSciNet): MR718069

Brown, G. and Yin, Q., Metrical theory for Farey continued fractions, Osaka J. Math., 33 (1996), no. 4, 951--970.

Mathematical Reviews (MathSciNet): MR1435463

Zentralblatt MATH: 0880.11057

Project Euclid: euclid.ojm/1200787226

Denjoy, A., Complément à la notice publiée en 1934 sur les travaux scientifiques de M. Arnaud Denjoy, Hermann, Paris, 1942.

Dajani, K. and Kraaikamp, C., The mother of all continued fractions, Colloq. Math., 84/85 (2000), 109--123.

Mathematical Reviews (MathSciNet): MR1778844

Zentralblatt MATH: 0961.11027

Foguel, S. R., The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969.

Mathematical Reviews (MathSciNet): MR261686

Zentralblatt MATH: 0282.60037

Iosifescu, Marius and Grigorescu, Serban, Dependence with Complete Connections and its Applications, Cambridge Tracts in Mathematics, 96. Cambridge Univ. Press, Cambridge, 1990.

Mathematical Reviews (MathSciNet): MR1070097

Iosifescu, Marius and Kraaikamp, Cor, Metrical Theory of Continued Fractions, Mathematics and its Applications, 547. Kluwer Academic Publishers, Dordrecht, 2002.

Mathematical Reviews (MathSciNet): MR1960327

Iosifescu, M. and Kraaikamp, C., On Denjoy's canonical continued fraction expansion, Osaka J. Math., 40 (2003), no. 1, 235--244.

Mathematical Reviews (MathSciNet): MR1955806

Zentralblatt MATH: 1046.11052

Project Euclid: euclid.ojm/1153493045

Ito, Sh., Algorithms with mediant convergents and their metrical theory, Osaka J. Math., 26 (1989), no. 3, 557--578.

Mathematical Reviews (MathSciNet): MR1021431

Zentralblatt MATH: 0702.11046

Project Euclid: euclid.ojm/1200781697

Kraaikamp, C., A new class of continued fraction expansions, Acta Arith., 57 (1991), no. 1, 1--39.

Mathematical Reviews (MathSciNet): MR1093246

Nakada, H., Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math., 4 (1981), no. 2, 399--426.

Mathematical Reviews (MathSciNet): MR646050

Zentralblatt MATH: 0479.10029

Nakada, Hitoshi, Ito, Shunji and Tanaka, Shigeru, On the invariant measure for the transformations associated with some real continued-fractions, Keio Engrg. Rep., 30 (1977), no. 13, 159--175.

Mathematical Reviews (MathSciNet): MR498461

Walters, P., An Introduction to Ergodic Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1982.

Mathematical Reviews (MathSciNet): MR648108

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