Metric theorem and Hausdorff dimension on recurrence rate of Laurent series
Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 157-171
Printed January 1, 2014
Xue-hai Hu, Bing Li, and Jian Xu
Huazhong Agricultural University, South China University of Technology, Huazhong University of Science and Technology
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 $E_{\alpha,\beta}$ be the set of points with lower and upper recurrence rates $\alpha$, $\beta$ ($0\leq \alpha \leq \beta\leq \infty$), we prove that all the sets $E_{\alpha,\beta}$ are of full Hausdorff dimension. Then the recurrence sets $E_{\alpha,\beta}$ have constant multifractal spectra.
Keywords : recurrence rate, pointwise dimension, continued fractions, Laurent series, Hausdorff dimension
MSC numbers : 28A80
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