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Construction of Lyapunov functions using Helmholtz–Hodge decomposition
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 |
$ (X, d) $ |
$ T:X\to X $ |
$ X $ |
$ \{a_n\} $ |
$ y\in X $ |
$ 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\}, $ |
$ B_n(x, r) $ |
$ 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}, $ |
$ (X, T) $ |
$ h_{\text {top}} $ |
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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