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A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates. (English summary) J. Reine Angew. Math. 605 (2007), 133–163. The continued fraction expansion x = [a1(x),...] and its sequence of associated approximants defined by pn(x)/qn(x) = [a1(x),..., an(x)] are generated by a uniformly hyperbolic dynamical system on an infinite alphabet or by a nonuniformly hyperbolic system on a finite alphabet. The usual theory of multifractals does not apply in either case. Classical results of P. Lévy [Bull. Soc. Math. France 57 (1929), 178–194; MR1504948; JFM 55.0916.02] and A. Ya. Khinchin [Compos. Math. 3 (1936), 276–285; Zbl 0014.25402] show that l1(x) = lim (2 log qn(x))/ n→ ∞ ( n ∑ (x) ) = 0, l2(x) = lim n→∞ i=1 ( n ∑ (x) ) /n = ∞ i=1 and l3(x) = lim (2 log qn(x))/n = C> 0 n→∞ almost everywhere. Defining the corresponding level sets by Li(s) = {x ∈ [0, 1) | li(x) = s}, it follows that dimH(L1(0)) = dimH(L2(∞) ∩ L3(C)) = 1. In the paper under review the relationship between these Hausdorff dimensions for certain nongeneric limiting behaviour is studied. An example of the results obtained is that for any α ∈ [0, 2 log γ] (γ the golden mean) there is another number α ′ = α ′(α) ∈ R ∪ {∞} for which dimH(L1(α)) = dimH(L2(α ′) ∩ L3(α · α ′)), where α ′(0) = ∞ and 0 · α ′(0) = C. The methods used involve the relation to the modular surface and intricate properties of associated pressure functions

Year: 2011

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