American Institute of Mathematical Sciences

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

Diophantine approximation of the orbits in topological dynamical systems

Chao Ma 1, , Baowei Wang 2, and  Jun Wu 2,,

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.
Keywords: Shrinking target, topological entropy, specification.
Mathematics Subject Classification: Primary: 11J83; Secondary: 28D20.
Citation: Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete and Continuous Dynamical Systems, 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.

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.

[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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