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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 and Continuous Dynamical Systems, 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, New York-London, 1997. doi: 10.1007/978-1-4757-2668-8.

[2]

L. Barreira and B. Saussol, Hausdoff dimension of measures via Poincaré Recurrence, Comm. Math. Phys., 219 (2001), 443-463. doi: 10.1007/s002200100427.

[3]

M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320.

[4]

M. Boshernitzan and E. Glasner, On two recurrence problems, Fund. Math., 206 (2009), 113-130. doi: 10.4064/fm206-0-7.

[5]

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

[6]

H. Fursternberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc., 5 (1981), 211-234. doi: 10.1090/S0273-0979-1981-14932-6.

[7]

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

[8]

E. Glasner, Classifying dynamical systems by their recurrence properties, Topol. Methods Nonlinear Anal., 24 (2004), 21-40.

[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), 603-627. doi: 10.1007/s10587-013-0043-z.

[10]

S. Grivaux, Non-recurrence sets for weakly mixing linear dynamical systems, Ergodic Theory Dynam. Systems, 34 (2014), 132-152. doi: 10.1017/etds.2012.116.

[11]

R. Hill and S. Velani, The shrinking target problems for matrix transformations of tori, J. London Math. Soc. (2), 60 (1999), 381-398. doi: 10.1112/S0024610799007681.

[12]

E. Manfred and W. Thomas, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.

[13]

B. Tan and B. W. Wang, Quantitative reccurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097. doi: 10.1016/j.aim.2011.06.034.

show all references

References:
[1]

E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, Plenum Press, New York-London, 1997. doi: 10.1007/978-1-4757-2668-8.

[2]

L. Barreira and B. Saussol, Hausdoff dimension of measures via Poincaré Recurrence, Comm. Math. Phys., 219 (2001), 443-463. doi: 10.1007/s002200100427.

[3]

M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320.

[4]

M. Boshernitzan and E. Glasner, On two recurrence problems, Fund. Math., 206 (2009), 113-130. doi: 10.4064/fm206-0-7.

[5]

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

[6]

H. Fursternberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc., 5 (1981), 211-234. doi: 10.1090/S0273-0979-1981-14932-6.

[7]

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

[8]

E. Glasner, Classifying dynamical systems by their recurrence properties, Topol. Methods Nonlinear Anal., 24 (2004), 21-40.

[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), 603-627. doi: 10.1007/s10587-013-0043-z.

[10]

S. Grivaux, Non-recurrence sets for weakly mixing linear dynamical systems, Ergodic Theory Dynam. Systems, 34 (2014), 132-152. doi: 10.1017/etds.2012.116.

[11]

R. Hill and S. Velani, The shrinking target problems for matrix transformations of tori, J. London Math. Soc. (2), 60 (1999), 381-398. doi: 10.1112/S0024610799007681.

[12]

E. Manfred and W. Thomas, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011. doi: 10.1007/978-0-85729-021-2.

[13]

B. Tan and B. W. Wang, Quantitative reccurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097. doi: 10.1016/j.aim.2011.06.034.

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