# METRIC THEOREM AND HAUSDORFF DIMENSION ON RECURRENCE RATE OF LAURENT SERIES

• Hu, Xue-Hai (College of Science Huazhong Agricultural University) ;
• Li, Bing (Department of Mathematics South China University of Technology) ;
• Xu, Jian (School of Mathematics and Statistics Huazhong University of Science and Technology)
• Published : 2014.01.31

#### 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.

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