# American Institute of Mathematical Sciences

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June  2016, 36(6): 2969-2979. doi: 10.3934/dcds.2016.36.2969

## Moving recurrent properties for the doubling map on the unit interval

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2 Faculty of Information Technology, Department of General Education, Macau University of Science and Technology, Macau 3 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan

Received  May 2015 Revised  September 2015 Published  December 2015

Let $(X,T,\mathcal{B}, \mu)$ be a measure-theoretical dynamical system with a compatible metric $d.$ Following Boshernitzan, call a point $x\in X$ is $\{n_{k}\}$-moving recurrent if $$\inf_{k\geq1} d\big(T^{n_{k}}x, \ T^{n_k+{k}}x\big)=0,$$ where $\{n_{k}\}_{k\in \mathbb{N}}$ is a given sequence of integers. It was asked whether the set of $\{n_{k}\}$-moving recurrent points is of full $\mu$-measure. In this paper, we restrict our attention to the doubling map and quantify the size of the set of $\{n_{k}\}$-moving recurrent points in the sense of measure (a class of $2$-fold mixing measures) and Hausdorff dimension.
Citation: Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969
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