METRIC THEOREM AND HAUSDORFF DIMENSION ON RECURRENCE RATE OF LAURENT SERIES
METRIC THEOREM AND HAUSDORFF DIMENSION ON RECURRENCE RATE OF LAURENT SERIES
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DOI：
10.4134/bkms.2014.51.1.157
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发表时间：
2014-01
影响因子：
0.5
通讯作者：
Xue-Hai Hu;Bing Li;Jian Xu
中科院分区：
文献类型：
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作者：
Xue-Hai Hu;Bing Li;Jian Xu
抽象的。我们证明了关于连分式的Laurent级数的常返率几乎必然与其Haar测度的逐点维数一致。此外，设E α，β是具有上、下递归率α，β（0 ≤ α ≤ β ≤ ∞）的点集，我们证明了E α，β是全Hausdor维数的.则递归集E α，β具有常数重分形谱. 1.设Fq是一个q元有限域，Fq（z-1）表示Fq中所有具系数的形式Laurent级数的域，Fq [z]表示Fq中具系数的多项式环，对任意x =P ∞n=n 0 cnz-n ∈ Fq（z-1），其中n 0 ∈ Z，记[x] =X 0 n =n 0 cnz-n ∈ Fq [z]，称之为x的整数部分，x的阶定义为v（x）=−deg（x）= inf{n ∈ Z：c n 6= 0 }。定义F q（z −1）上的一个非阿基米德赋值为kxk = q −v（x），对所有x ∈ F q（z −1）。在度量ρ（x，x ′）= kx −x ′ k下，场F q（z −1）是局部紧的和完备的。由于赋值k·k是非阿基米德的，因此，如果两个圆盘相交，则其中一个包含另一个。
Abstract. We show that the recurrence rates of Laurent series aboutcontinued fractions almost surely coincide with their pointwise dimensionsof the Haar measure. Moreover, let E α,β be the set of points with lowerand upper recurrence rates α, β (0 ≤ α ≤ β ≤ ∞), we prove that all thesets E α,β are of full Hausdorff dimension. Then the recurrence sets E α,β have constant multifractal spectra. 1. IntroductionLet F q be a finite field of q elements and F q (z −1 )denote the field ofall formal Laurent series with coefficients in F q , and F q [z] denote the ring ofpolynomials with coefficients in F q .For each x =P ∞n=n 0 c n z −n ∈ F q (z −1 )with n 0 ∈ Z, denote[x] =X 0n=n 0 c n z −n ∈ F q [z],which is called the integer part of x, and the order of x is defined as v(x) =−deg(x) = inf{n ∈ Z : c n 6= 0 }. Define a non-Archimedean valuation onF q (z −1 )as kxk = q −v(x) for all x ∈ F q (z −1 ). The field F q (z −1 )islocally compact and complete under the metric ρ(x,x ′ ) = kx −x ′ k.Remark 1.1. Since the valuation k·k is non-Archimedean, it follows that if twodiscs intersect, then one contains the other.Let I denote the valuation ideal of F