Continuity of the Hausdorff Measure of Continued Fractions and Countable Alphabet Iterated Function Systems

Urbański, Mariusz; Zdunik, Anna

Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 261-286

We prove that if by $J_n\left(𝒢\right)$ 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 ${lim}_{n\to \infty }{\text{H}}_{h_n}\left(J_n\left(𝒢\right)\right)=1={\text{H}}_1\left(J\left(𝒢\right)\right)$, where $h_n$ is the Hausdorff dimension of $J_n\left(𝒢\right)$, ${\text{H}}_{h_n}$ is the corresponding Hausdorff measure, and $J\left(𝒢\right)$ 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 ${\underline{lim}}_{F\to E}{\text{H}}_{h_F}\left(J_F\right)<{\text{H}}_{h_{𝒮}}\left(J_{𝒮}\right)$; the lower limit is taken over finite subsets of the countable infinite alphabet $E$.

Nous montrons que, si $J_n\left(𝒢\right)$ est l’ensemble des réels dans $[0,1]$ dont la fraction continue infinie est constituée de nombres entiers compris entre $1$ et $n$, alors ${lim}_{n\to \infty }{H}_{h_n}\left(J_n\left(𝒢\right)\right)=1={\text{H}}_1\left(J\left(𝒢\right)\right)$, où $h_n$ est la dimension de Hausdorff de $J_n\left(𝒢\right)$, ${\text{H}}_{h_n}$ est la mesure de Hausdorff correspondant et où $J\left(𝒢\right)$ 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 ${\underline{lim}}_{F\to E}{\text{H}}_{h_F}\left(J_F\right)<{\text{H}}_{h_{𝒮}}\left(J_{𝒮}\right)$ ; cette limite inférieure s’étend sur les sous-ensembles finis de l’alphabet infini $E$.

Classification : 11A55, 28A78, 28A80, 37D35
Keywords: Continued fractions, Hausdorff measure, Gauss map, bounded distortion, iterated function systems

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