Moving recurrent properties for the doubling map on the unit interval

Yuanhong  Chen; Chao  Ma; Jun Wu

doi:10.3934/dcds.2016.36.2969

Article Contents

2016, Volume 36, Issue 6: 2969-2979. Doi: 10.3934/dcds.2016.36.2969

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

Received: April 30, 2015

Revised: August 31, 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 11K55, 37D20; Secondary: 28A80.

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