Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems
Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 261-286.

Nous montrons que, si J n (𝒢) est l’ensemble des réels dans [0,1] dont la fraction continue infinie est constituée de nombres entiers compris entre 1 et n, alors lim n H h n (J n (𝒢))=1=H 1 (J(𝒢)), où h n est la dimension de Hausdorff de J n (𝒢), H h n est la mesure de Hausdorff correspondant et où J(𝒢) est l’ensemble de tous les nombres irrationnels de [0,1], i.e. ceux dont la fraction continue est infinie. Nous montrons aussi que cette propriété n’est pas générale en construisant une classe de systèmes de fonctions itérées 𝒮 sur [0,1], formés de similarités, pour lesquels lim ̲ FE H h F (J F )<H h 𝒮 (J 𝒮 ) ; cette limite inférieure s’étend sur les sous-ensembles finis de l’alphabet infini E.

We prove that if by J n (𝒢) we denote the set of all numbers in [0,1] whose infinite continued fraction expansions have all entries in the finite set {1,2,...,n}, then lim n H h n (J n (𝒢))=1=H 1 (J(𝒢)), where h n is the Hausdorff dimension of J n (𝒢), H h n is the corresponding Hausdorff measure, and J(𝒢) denotes the set of all irrational numbers in [0,1], i .e. those whose continued fraction expansion is infinite. We also show that this property is not too common by constructing a class of infinite iterated function systems 𝒮 on [0,1], consisting of similarities, for which lim ̲ FE H h F (J F )<H h 𝒮 (J 𝒮 ); the lower limit is taken over finite subsets of the countable infinite alphabet E.

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DOI : 10.5802/jtnb.938
Classification : 11A55, 28A78, 28A80, 37D35
Mots clés : Continued fractions, Hausdorff measure, Gauss map, bounded distortion, iterated function systems
Mariusz Urbański 1 ; Anna Zdunik 2

1 Department of Mathematics University of North Texas Denton, TX 76203-1430 USA
2 Institute of Mathematics University of Warsaw ul. Banacha 2, 02-097 Warszawa POLAND
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Mariusz Urbański; Anna Zdunik. Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems. Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 261-286. doi : 10.5802/jtnb.938. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.938/

[1] D. Hensley, « Continued fraction Cantor sets, Hausdorff dimension, and functional analysis », J. Number Theory 40 (1992), no. 3, p. 336-358. | DOI | MR | Zbl

[2] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995, Fractals and rectifiability, xii+343 pages. | DOI | Zbl

[3] R. D. Mauldin & M. Urbański, « Dimensions and measures in infinite iterated function systems », Proc. London Math. Soc. (3) 73 (1996), no. 1, p. 105-154. | DOI | MR | Zbl

[4] —, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003, Geometry and dynamics of limit sets, xii+281 pages. | DOI

[5] L. Olsen, « Hausdorff and packing measure functions of self-similar sets: continuity and measurability », Ergodic Theory Dynam. Systems 28 (2008), no. 5, p. 1635-1655. | DOI | MR | Zbl

[6] T. Szarek, M. Urbański & A. Zdunik, « Continuity of Hausdorff measure for conformal dynamical systems », Discrete Contin. Dyn. Syst. 33 (2013), no. 10, p. 4647-4692. | DOI | Zbl

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