# American Institute of Mathematical Sciences

May 2019, 39(5): 2455-2471. doi: 10.3934/dcds.2019104

## Diophantine approximation of the orbits in topological dynamical systems

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

* Corresponding author: Jun Wu

Received  January 2018 Revised  October 2018 Published  January 2019

Fund Project: This work is supported by the Science and Technology Development Fund of Macau (No. 044/2015/A2 and 0024/2018/A1) and National Nature Science Foundation of China (No. 11471130, 11722105, 11831007)

We would like to present a general principle for the shrinking target problem in a topological dynamical system. More precisely, let
 $(X, d)$
be a compact metric space and
 $T:X\to X$
a continuous transformation on
 $X$
. For any integer valued sequence
 $\{a_n\}$
and
 $y\in X$
, define
 $E_y(\{a_n\}) = \bigcap\limits_{\delta>0}\Big\{x\in X: T^nx\in B_{a_n}(y, \delta), \ {\text{for infinitely often}}\ n\in \mathbb N\Big\},$
the set of points whose orbit can well approximate a given point infinitely often, where
 $B_n(x, r)$
denotes the Bowen-ball. It is shown that
 $h_{\text {top}}(E_y(\{a_n\}), T) = \frac{1}{1+a}h_{\text {top}}(X, T), \ \ {\text{with}}\ a = \liminf\limits_{n\to\infty}\frac{a_n}{n},$
if the system
 $(X, T)$
has the specification property. Here
 $h_{\text {top}}$
denotes the topological entropy. An example is also given to indicate that the specification property required in the above result cannot be weakened even to almost specification.
Citation: Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104
##### References:
 [1] V. Beresnevich, D. Dickinson and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc., 179 (2006), ⅹ+91 pp. doi: 10.1090/memo/0846. [2] V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2), 164 (2006), 971–992. doi: 10.4007/annals.2006.164.971. [3] A. M. Blokh, Decomposition of dynamical systems on an interval, Usp. Mat. Nauk., 38 (1983), 179-180. [4] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [5] Y. Bugeaud and B. Wang, Distribution of full cylinders and the Diophantine properties of the orbits in β-expansions, J. Fractal Geom., 1 (2014), 221-241. doi: 10.4171/JFG/6. [6] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. [7] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemma for Gibbs measures, Mem. Amer. Math. Soc., 122 (2001), 1-27. doi: 10.1007/BF02809888. [8] A. Fan, L. Liao and J. Peyrière, Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dynam. Syst., 21 (2008), 1103-1128. doi: 10.3934/dcds.2008.21.1103. [9] A. Fan, J. Schemling and S. Troubetzkoy, A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation, Proc. London Math. Soc., 107 (2013), 1173-1219. doi: 10.1112/plms/pdt005. [10] S. Galatolo, Dimension via waiting time and recurrence, Math. Res. Lett., 12 (2005), 377-386. doi: 10.4310/MRL.2005.v12.n3.a8. [11] S. Galatolo and D. Kim, The dynamical Borel-Cantelli lemma and the waiting time problems, Indag. Math., 18 (2007), 421-434. doi: 10.1016/S0019-3577(07)80031-0. [12] R. Hill and S. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198. doi: 10.1007/BF01245179. [13] R. Hill and S. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Etudes Sci. Publ. Math., 85 (1997), 193-216. [14] D. Kim, The shrinking target property of irrational rotations, Nonlinearity, 20 (2007), 1637-1643. doi: 10.1088/0951-7715/20/7/006. [15] B. Li, B. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. (3), 108 (2014), 159–186. doi: 10.1112/plms/pdt017. [16] L. Liao and S. Seuret, Diophantine approximation by orbits in expanding Markov maps, Ergodic Th. Dynam. Systems, 33 (2013), 585-608. doi: 10.1017/S0143385711001039. [17] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hunger., 11 (1960), 401-416. doi: 10.1007/BF02020954. [18] Y. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [19] C.-E. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts, Nonlinearity, 18 (2005), 237-261. doi: 10.1088/0951-7715/18/1/013. [20] H. Revee, Shrinking targets for countable Markov maps, arXiv: 1107.4736 [21] J. Schmeling, Symbolic dynamics for β-shfits and self-normal numbers, Ergod. Th. Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. [22] L. Shen and B. Wang, Shrinking target problems for beta-dynamical system, Sci. China Math., 56 (2013), 91-104. doi: 10.1007/s11425-012-4478-8. [23] K. Sigmund, On dynamical systems with the specification property, Trans Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X. [24] B. Stratmann and M. Urbański, Jarník and Julia: A Diophantine analysis for parabolic rational maps, Math. Scand., 91 (2002), 27-54. doi: 10.7146/math.scand.a-14377. [25] F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergod. Th. Dynam. Systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. [26] D. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1. [27] M. Urbański, Diophantine analysis of conformal iterated function systems, Monatsh. Math., 137 (2002), 325-340. doi: 10.1007/s00605-002-0483-2. [28] B. Wang and J. Wu, A survey on the dimensional theory in dynamical Diophantine approximation, in Recent Developments in Fractals and Related Fields, Trends Math., Birkhäuser/Springer, Cham, (2017), 261–294. [29] C. Zhao and E. Chen, Quantitative recurrence properties for systems with non-uniform structure, Taiwanese J. Math., 22 (2018), 225-244. doi: 10.11650/tjm/8071.

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##### References:
 [1] V. Beresnevich, D. Dickinson and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc., 179 (2006), ⅹ+91 pp. doi: 10.1090/memo/0846. [2] V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2), 164 (2006), 971–992. doi: 10.4007/annals.2006.164.971. [3] A. M. Blokh, Decomposition of dynamical systems on an interval, Usp. Mat. Nauk., 38 (1983), 179-180. [4] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [5] Y. Bugeaud and B. Wang, Distribution of full cylinders and the Diophantine properties of the orbits in β-expansions, J. Fractal Geom., 1 (2014), 221-241. doi: 10.4171/JFG/6. [6] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. [7] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemma for Gibbs measures, Mem. Amer. Math. Soc., 122 (2001), 1-27. doi: 10.1007/BF02809888. [8] A. Fan, L. Liao and J. Peyrière, Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dynam. Syst., 21 (2008), 1103-1128. doi: 10.3934/dcds.2008.21.1103. [9] A. Fan, J. Schemling and S. Troubetzkoy, A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation, Proc. London Math. Soc., 107 (2013), 1173-1219. doi: 10.1112/plms/pdt005. [10] S. Galatolo, Dimension via waiting time and recurrence, Math. Res. Lett., 12 (2005), 377-386. doi: 10.4310/MRL.2005.v12.n3.a8. [11] S. Galatolo and D. Kim, The dynamical Borel-Cantelli lemma and the waiting time problems, Indag. Math., 18 (2007), 421-434. doi: 10.1016/S0019-3577(07)80031-0. [12] R. Hill and S. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198. doi: 10.1007/BF01245179. [13] R. Hill and S. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Etudes Sci. Publ. Math., 85 (1997), 193-216. [14] D. Kim, The shrinking target property of irrational rotations, Nonlinearity, 20 (2007), 1637-1643. doi: 10.1088/0951-7715/20/7/006. [15] B. Li, B. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. (3), 108 (2014), 159–186. doi: 10.1112/plms/pdt017. [16] L. Liao and S. Seuret, Diophantine approximation by orbits in expanding Markov maps, Ergodic Th. Dynam. Systems, 33 (2013), 585-608. doi: 10.1017/S0143385711001039. [17] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hunger., 11 (1960), 401-416. doi: 10.1007/BF02020954. [18] Y. B. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, University of Chicago Press, Chicago, 1997. doi: 10.7208/chicago/9780226662237.001.0001. [19] C.-E. Pfister and W. Sullivan, Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts, Nonlinearity, 18 (2005), 237-261. doi: 10.1088/0951-7715/18/1/013. [20] H. Revee, Shrinking targets for countable Markov maps, arXiv: 1107.4736 [21] J. Schmeling, Symbolic dynamics for β-shfits and self-normal numbers, Ergod. Th. Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. [22] L. Shen and B. Wang, Shrinking target problems for beta-dynamical system, Sci. China Math., 56 (2013), 91-104. doi: 10.1007/s11425-012-4478-8. [23] K. Sigmund, On dynamical systems with the specification property, Trans Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X. [24] B. Stratmann and M. Urbański, Jarník and Julia: A Diophantine analysis for parabolic rational maps, Math. Scand., 91 (2002), 27-54. doi: 10.7146/math.scand.a-14377. [25] F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergod. Th. Dynam. Systems, 23 (2003), 317-348. doi: 10.1017/S0143385702000913. [26] D. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1. [27] M. Urbański, Diophantine analysis of conformal iterated function systems, Monatsh. Math., 137 (2002), 325-340. doi: 10.1007/s00605-002-0483-2. [28] B. Wang and J. Wu, A survey on the dimensional theory in dynamical Diophantine approximation, in Recent Developments in Fractals and Related Fields, Trends Math., Birkhäuser/Springer, Cham, (2017), 261–294. [29] C. Zhao and E. Chen, Quantitative recurrence properties for systems with non-uniform structure, Taiwanese J. Math., 22 (2018), 225-244. doi: 10.11650/tjm/8071.
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