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Moving recurrent properties for the doubling map on the unit interval

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


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