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Metric theorem and Hausdorff dimension on recurrence rate of Laurent series

Bing Li|Jian Xu|Xuehai Hu
대한수학회 (The Korean Mathematical Society)|2014.01|대한수학회보 (Bulletin of the KMS)|제51권 제1호|pp. 157-171

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참고문헌

키워드

Hausdorff dimensionLaurent seriescontinued fractionsrecurrence ratepointwise dimension
주제분류수학

초록

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let Eα,β be the set of points with lower and upper recurrence rates α, (0 ≤ α ≤ β≤ 1), we prove that all the sets Eα,β are of full Hausdorff dimension. Then the recurrence sets Eα,β have constant multifractal spectra.

Abstract

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let Eα,β be the set of points with lower and upper recurrence rates α, (0 ≤ α ≤ β ≤ 1), we prove that all the sets Eα,β are of full Hausdorff dimension. Then the recurrence sets Eα,β have constant multifractal spectra.

참고문헌 (16)

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