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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
References:
[1]

E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions,, Plenum Press, (1997). doi: 10.1007/978-1-4757-2668-8. Google Scholar

[2]

L. Barreira and B. Saussol, Hausdoff dimension of measures via Poincaré Recurrence,, Comm. Math. Phys., 219 (2001), 443. doi: 10.1007/s002200100427. Google Scholar

[3]

M. Boshernitzan, Quantitative recurrence results,, Invent. Math., 113 (1993), 617. doi: 10.1007/BF01244320. Google Scholar

[4]

M. Boshernitzan and E. Glasner, On two recurrence problems,, Fund. Math., 206 (2009), 113. doi: 10.4064/fm206-0-7. Google Scholar

[5]

K. J. Falconer, Fractal Geometry, Mathematical Foundations and Application,, John Wiley and Sons, (1990). doi: 10.1002/0470013850. Google Scholar

[6]

H. Fursternberg, Poincaré recurrence and number theory,, Bull. Amer. Math. Soc., 5 (1981), 211. doi: 10.1090/S0273-0979-1981-14932-6. Google Scholar

[7]

H. Fursternberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton university press, (1981). doi: 10.1090/s0273-0979-1986-15451-0. Google Scholar

[8]

E. Glasner, Classifying dynamical systems by their recurrence properties,, Topol. Methods Nonlinear Anal., 24 (2004), 21. Google Scholar

[9]

S. Grivaux and M. Roginskaya, Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle,, Czechoslovak Math. J., 63 (2013), 603. doi: 10.1007/s10587-013-0043-z. Google Scholar

[10]

S. Grivaux, Non-recurrence sets for weakly mixing linear dynamical systems,, Ergodic Theory Dynam. Systems, 34 (2014), 132. doi: 10.1017/etds.2012.116. Google Scholar

[11]

R. Hill and S. Velani, The shrinking target problems for matrix transformations of tori,, J. London Math. Soc. (2), 60 (1999), 381. doi: 10.1112/S0024610799007681. Google Scholar

[12]

E. Manfred and W. Thomas, Ergodic Theory with a View Towards Number Theory,, Graduate Texts in Mathematics, (2011). doi: 10.1007/978-0-85729-021-2. Google Scholar

[13]

B. Tan and B. W. Wang, Quantitative reccurrence properties for beta-dynamical system,, Adv. Math., 228 (2011), 2071. doi: 10.1016/j.aim.2011.06.034. Google Scholar

show all references

References:
[1]

E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions,, Plenum Press, (1997). doi: 10.1007/978-1-4757-2668-8. Google Scholar

[2]

L. Barreira and B. Saussol, Hausdoff dimension of measures via Poincaré Recurrence,, Comm. Math. Phys., 219 (2001), 443. doi: 10.1007/s002200100427. Google Scholar

[3]

M. Boshernitzan, Quantitative recurrence results,, Invent. Math., 113 (1993), 617. doi: 10.1007/BF01244320. Google Scholar

[4]

M. Boshernitzan and E. Glasner, On two recurrence problems,, Fund. Math., 206 (2009), 113. doi: 10.4064/fm206-0-7. Google Scholar

[5]

K. J. Falconer, Fractal Geometry, Mathematical Foundations and Application,, John Wiley and Sons, (1990). doi: 10.1002/0470013850. Google Scholar

[6]

H. Fursternberg, Poincaré recurrence and number theory,, Bull. Amer. Math. Soc., 5 (1981), 211. doi: 10.1090/S0273-0979-1981-14932-6. Google Scholar

[7]

H. Fursternberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton university press, (1981). doi: 10.1090/s0273-0979-1986-15451-0. Google Scholar

[8]

E. Glasner, Classifying dynamical systems by their recurrence properties,, Topol. Methods Nonlinear Anal., 24 (2004), 21. Google Scholar

[9]

S. Grivaux and M. Roginskaya, Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle,, Czechoslovak Math. J., 63 (2013), 603. doi: 10.1007/s10587-013-0043-z. Google Scholar

[10]

S. Grivaux, Non-recurrence sets for weakly mixing linear dynamical systems,, Ergodic Theory Dynam. Systems, 34 (2014), 132. doi: 10.1017/etds.2012.116. Google Scholar

[11]

R. Hill and S. Velani, The shrinking target problems for matrix transformations of tori,, J. London Math. Soc. (2), 60 (1999), 381. doi: 10.1112/S0024610799007681. Google Scholar

[12]

E. Manfred and W. Thomas, Ergodic Theory with a View Towards Number Theory,, Graduate Texts in Mathematics, (2011). doi: 10.1007/978-0-85729-021-2. Google Scholar

[13]

B. Tan and B. W. Wang, Quantitative reccurrence properties for beta-dynamical system,, Adv. Math., 228 (2011), 2071. doi: 10.1016/j.aim.2011.06.034. Google Scholar

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