-
Previous Article
Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium
- DCDS Home
- This Issue
-
Next Article
On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations
On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property
1. | National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic |
2. | National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic, – and –, AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland |
3. | School of Mathematical Science, Fudan University, Shanghai 200433, China |
We study the properties of $ \Phi $-irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of $ \Phi $-irregular set in terms of entropy on chain recurrent classes and prove that $ \Phi $-irregular sets of full entropy are typical. We also consider $ \Phi $-level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.
References:
[1] |
L. Barreira and J. Schmeling,
Sets of non-typical points have full topological measure and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.
doi: 10.1007/BF02773211. |
[2] |
R. Bowen,
Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.
doi: 10.1090/S0002-9947-1972-0285689-X. |
[3] |
R. Bowen,
Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
doi: 10.2307/1995452. |
[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] |
M. Brin and A. Katok,
On local entropy, Geom. Dyn. Springer Lecture Notes, 1007 (1983), 30-38.
doi: 10.1007/BFb0061408. |
[6] |
L. Chen,
Linking and the shadowing property for piecewise monotone maps, Proc. Amer. Math. Soc., 113 (1991), 251-263.
doi: 10.1090/S0002-9939-1991-1079695-2. |
[7] |
E. M. Coven, I. Kan and J. A. Yorke,
Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988), 227-241.
doi: 10.1090/S0002-9947-1988-0946440-2. |
[8] |
R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. ![]() ![]() |
[9] |
Y. Dong, P. Oprocha and X. Tian, On the irregular points for systems with the shadowing property, Ergodic Theory Dynam. Systems, 38 (2018), 2108–2131.
doi: 10.1017/etds.2016.126. |
[10] |
C. Ercai, T. Küpper and S. Lin,
Topological entropy for divergence points, Ergod. Th. Dyn. Sys., 25 (2005), 1173-1208.
doi: 10.1017/S0143385704000872. |
[11] |
A.-H. Fan, D.-J. Feng and J. Wu,
Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.
doi: 10.1017/S0024610701002137. |
[12] |
C. Good and J. Meddaugh, Shifts of finite type as fundamental objects in the theory of shadowing, Invent. Math., 220 (2020), 715–736. arXiv: 1702.05170.
doi: 10.1007/s00222-019-00936-8. |
[13] |
M. W. Hirsch, H. L. Smith and X.-Q. Zhao,
Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Diff. Eq., 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
[14] |
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-174.
|
[15] |
J. Li and P. Oprocha,
Properties of invariant measures in dynamical systems with the shadowing property, Erg. Theory and Dyn. Sys., 38 (2018), 2257-2294.
doi: 10.1017/etds.2016.125. |
[16] |
J. Li and M. Wu,
Generic property of irregular sets in systems satisfying the specification property, Discrete Contin. Dyn. Sys., 34 (2014), 635-645.
doi: 10.3934/dcds.2014.34.635. |
[17] |
T. K. S. Moothathu,
Implications of pseudo-orbit tracing property of continuous maps on compacta, Top. Appl., 158 (2011), 2232-2239.
doi: 10.1016/j.topol.2011.07.016. |
[18] |
T. K. S. Moothathu and P. Oprocha,
Shadowing, entropy and minimal subsystems, Monatsh. Math., 172 (2013), 357-378.
doi: 10.1007/s00605-013-0504-3. |
[19] |
L. Olsen,
Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713.
doi: 10.4310/MRL.2002.v9.n6.a1. |
[20] |
L. Olsen,
Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.
doi: 10.1016/j.matpur.2003.09.007. |
[21] |
L. Olsen and S. Winter,
Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc., 67 (2003), 103-122.
doi: 10.1112/S0024610702003630. |
[22] |
D. Ruelle, Historic behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, 63–66, Inst. Phys., Bristol, 2001. |
[23] |
K. Sigmund,
Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.
doi: 10.1007/BF01404606. |
[24] |
F. Takens and E. Verbitskiy,
On the variational principle for the topological entropy of certain non-compact sets, Erg. Theory Dynam. Sys., 23 (2003), 317-348.
doi: 10.1017/S0143385702000913. |
[25] |
X. Tian,
Topological pressure for the completely irregular set of Birkhoff averages, Discrete Contin. Dyn. Sys., 37 (2017), 2745-2763.
doi: 10.3934/dcds.2017118. |
[26] |
D. J. Thompson,
Irregular sets, the $\beta$-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414.
doi: 10.1090/S0002-9947-2012-05540-1. |
[27] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[28] |
L.-S. Young,
Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.
doi: 10.2307/2001318. |
show all references
References:
[1] |
L. Barreira and J. Schmeling,
Sets of non-typical points have full topological measure and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70.
doi: 10.1007/BF02773211. |
[2] |
R. Bowen,
Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.
doi: 10.1090/S0002-9947-1972-0285689-X. |
[3] |
R. Bowen,
Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.
doi: 10.2307/1995452. |
[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] |
M. Brin and A. Katok,
On local entropy, Geom. Dyn. Springer Lecture Notes, 1007 (1983), 30-38.
doi: 10.1007/BFb0061408. |
[6] |
L. Chen,
Linking and the shadowing property for piecewise monotone maps, Proc. Amer. Math. Soc., 113 (1991), 251-263.
doi: 10.1090/S0002-9939-1991-1079695-2. |
[7] |
E. M. Coven, I. Kan and J. A. Yorke,
Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988), 227-241.
doi: 10.1090/S0002-9947-1988-0946440-2. |
[8] |
R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. ![]() ![]() |
[9] |
Y. Dong, P. Oprocha and X. Tian, On the irregular points for systems with the shadowing property, Ergodic Theory Dynam. Systems, 38 (2018), 2108–2131.
doi: 10.1017/etds.2016.126. |
[10] |
C. Ercai, T. Küpper and S. Lin,
Topological entropy for divergence points, Ergod. Th. Dyn. Sys., 25 (2005), 1173-1208.
doi: 10.1017/S0143385704000872. |
[11] |
A.-H. Fan, D.-J. Feng and J. Wu,
Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.
doi: 10.1017/S0024610701002137. |
[12] |
C. Good and J. Meddaugh, Shifts of finite type as fundamental objects in the theory of shadowing, Invent. Math., 220 (2020), 715–736. arXiv: 1702.05170.
doi: 10.1007/s00222-019-00936-8. |
[13] |
M. W. Hirsch, H. L. Smith and X.-Q. Zhao,
Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dynam. Diff. Eq., 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
[14] |
A. Katok,
Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-174.
|
[15] |
J. Li and P. Oprocha,
Properties of invariant measures in dynamical systems with the shadowing property, Erg. Theory and Dyn. Sys., 38 (2018), 2257-2294.
doi: 10.1017/etds.2016.125. |
[16] |
J. Li and M. Wu,
Generic property of irregular sets in systems satisfying the specification property, Discrete Contin. Dyn. Sys., 34 (2014), 635-645.
doi: 10.3934/dcds.2014.34.635. |
[17] |
T. K. S. Moothathu,
Implications of pseudo-orbit tracing property of continuous maps on compacta, Top. Appl., 158 (2011), 2232-2239.
doi: 10.1016/j.topol.2011.07.016. |
[18] |
T. K. S. Moothathu and P. Oprocha,
Shadowing, entropy and minimal subsystems, Monatsh. Math., 172 (2013), 357-378.
doi: 10.1007/s00605-013-0504-3. |
[19] |
L. Olsen,
Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713.
doi: 10.4310/MRL.2002.v9.n6.a1. |
[20] |
L. Olsen,
Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl., 82 (2003), 1591-1649.
doi: 10.1016/j.matpur.2003.09.007. |
[21] |
L. Olsen and S. Winter,
Normal and non-normal points of self-similar sets and divergence points of self-similar measures, J. London Math. Soc., 67 (2003), 103-122.
doi: 10.1112/S0024610702003630. |
[22] |
D. Ruelle, Historic behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, 63–66, Inst. Phys., Bristol, 2001. |
[23] |
K. Sigmund,
Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.
doi: 10.1007/BF01404606. |
[24] |
F. Takens and E. Verbitskiy,
On the variational principle for the topological entropy of certain non-compact sets, Erg. Theory Dynam. Sys., 23 (2003), 317-348.
doi: 10.1017/S0143385702000913. |
[25] |
X. Tian,
Topological pressure for the completely irregular set of Birkhoff averages, Discrete Contin. Dyn. Sys., 37 (2017), 2745-2763.
doi: 10.3934/dcds.2017118. |
[26] |
D. J. Thompson,
Irregular sets, the $\beta$-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414.
doi: 10.1090/S0002-9947-2012-05540-1. |
[27] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[28] |
L.-S. Young,
Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.
doi: 10.2307/2001318. |
[1] |
Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 |
[2] |
Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062 |
[3] |
Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365 |
[4] |
Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328 |
[5] |
Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278 |
[6] |
Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 |
[7] |
Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 |
[8] |
Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293 |
[9] |
Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228 |
[10] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020378 |
[11] |
Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265 |
[12] |
Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067 |
[13] |
Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020123 |
[14] |
Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 |
[15] |
Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 |
[16] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020447 |
[17] |
Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020445 |
[18] |
Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020442 |
[19] |
Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020440 |
[20] |
Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]