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. Google Scholar

[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. Google Scholar

[3]

A. M. Blokh, Decomposition of dynamical systems on an interval, Usp. Mat. Nauk., 38 (1983), 179-180. Google Scholar

[4]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar

[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. Google Scholar

[6]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. Google Scholar

[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. Google Scholar

[8]

A. FanL. 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. Google Scholar

[9]

A. FanJ. 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. Google Scholar

[10]

S. Galatolo, Dimension via waiting time and recurrence, Math. Res. Lett., 12 (2005), 377-386. doi: 10.4310/MRL.2005.v12.n3.a8. Google Scholar

[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. Google Scholar

[12]

R. Hill and S. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198. doi: 10.1007/BF01245179. Google Scholar

[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. Google Scholar

[14]

D. Kim, The shrinking target property of irrational rotations, Nonlinearity, 20 (2007), 1637-1643. doi: 10.1088/0951-7715/20/7/006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hunger., 11 (1960), 401-416. doi: 10.1007/BF02020954. Google Scholar

[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. Google Scholar
[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. Google Scholar

[20]

H. Revee, Shrinking targets for countable Markov maps, arXiv: 1107.4736Google Scholar

[21]

J. Schmeling, Symbolic dynamics for β-shfits and self-normal numbers, Ergod. Th. Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

M. Urbański, Diophantine analysis of conformal iterated function systems, Monatsh. Math., 137 (2002), 325-340. doi: 10.1007/s00605-002-0483-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

A. M. Blokh, Decomposition of dynamical systems on an interval, Usp. Mat. Nauk., 38 (1983), 179-180. Google Scholar

[4]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. Google Scholar

[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. Google Scholar

[6]

J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. Google Scholar

[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. Google Scholar

[8]

A. FanL. 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. Google Scholar

[9]

A. FanJ. 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. Google Scholar

[10]

S. Galatolo, Dimension via waiting time and recurrence, Math. Res. Lett., 12 (2005), 377-386. doi: 10.4310/MRL.2005.v12.n3.a8. Google Scholar

[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. Google Scholar

[12]

R. Hill and S. Velani, The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198. doi: 10.1007/BF01245179. Google Scholar

[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. Google Scholar

[14]

D. Kim, The shrinking target property of irrational rotations, Nonlinearity, 20 (2007), 1637-1643. doi: 10.1088/0951-7715/20/7/006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hunger., 11 (1960), 401-416. doi: 10.1007/BF02020954. Google Scholar

[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. Google Scholar
[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. Google Scholar

[20]

H. Revee, Shrinking targets for countable Markov maps, arXiv: 1107.4736Google Scholar

[21]

J. Schmeling, Symbolic dynamics for β-shfits and self-normal numbers, Ergod. Th. Dynam. Systems, 17 (1997), 675-694. doi: 10.1017/S0143385797079182. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

M. Urbański, Diophantine analysis of conformal iterated function systems, Monatsh. Math., 137 (2002), 325-340. doi: 10.1007/s00605-002-0483-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

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