doi: 10.3934/dcds.2020317

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

* Corresponding author: oprocha@agh.edu.pl

Received  November 2019 Revised  May 2020 Published  August 2020

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.

Citation: Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020317
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.  Google Scholar

[2]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  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]

M. Brin and A. Katok, On local entropy, Geom. Dyn. Springer Lecture Notes, 1007 (1983), 30-38.  doi: 10.1007/BFb0061408.  Google Scholar

[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.  Google Scholar

[7]

E. M. CovenI. 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.  Google Scholar

[8] R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Adv. Math, 74. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755347.  Google Scholar
[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.  Google Scholar

[10]

C. ErcaiT. Küpper and S. Lin, Topological entropy for divergence points, Ergod. Th. Dyn. Sys., 25 (2005), 1173-1208.  doi: 10.1017/S0143385704000872.  Google Scholar

[11]

A.-H. FanD.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[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.  Google Scholar

[13]

M. W. HirschH. 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.  Google Scholar

[14]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-174.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Olsen, Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713.  doi: 10.4310/MRL.2002.v9.n6.a1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Ruelle, Historic behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, 63–66, Inst. Phys., Bristol, 2001.  Google Scholar

[23]

K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[28]

L.-S. Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.  doi: 10.2307/2001318.  Google Scholar

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.  Google Scholar

[2]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom $A$ diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  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]

M. Brin and A. Katok, On local entropy, Geom. Dyn. Springer Lecture Notes, 1007 (1983), 30-38.  doi: 10.1007/BFb0061408.  Google Scholar

[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.  Google Scholar

[7]

E. M. CovenI. 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.  Google Scholar

[8] R. M. Dudley, Real Analysis and Probability, Revised reprint of the 1989 original. Cambridge Studies in Adv. Math, 74. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755347.  Google Scholar
[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.  Google Scholar

[10]

C. ErcaiT. Küpper and S. Lin, Topological entropy for divergence points, Ergod. Th. Dyn. Sys., 25 (2005), 1173-1208.  doi: 10.1017/S0143385704000872.  Google Scholar

[11]

A.-H. FanD.-J. Feng and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc., 64 (2001), 229-244.  doi: 10.1017/S0024610701002137.  Google Scholar

[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.  Google Scholar

[13]

M. W. HirschH. 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.  Google Scholar

[14]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-174.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

L. Olsen, Divergence points of deformed empirical measures, Math. Res. Lett., 9 (2002), 701-713.  doi: 10.4310/MRL.2002.v9.n6.a1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

D. Ruelle, Historic behaviour in smooth dynamical systems, Global Analysis of Dynamical Systems, 63–66, Inst. Phys., Bristol, 2001.  Google Scholar

[23]

K. Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[28]

L.-S. Young, Some large deviation results for dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543.  doi: 10.2307/2001318.  Google Scholar

[1]

Thomas French. Follower, predecessor, and extender set sequences of $ \beta $-shifts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4331-4344. doi: 10.3934/dcds.2019175

[2]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020267

[3]

Jiaoxiu Ling, Zhan Zhou. Positive solutions of the discrete Robin problem with $ \phi $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020338

[4]

Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $ \phi $-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082

[5]

Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2085-2095. doi: 10.3934/dcdss.2019134

[6]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020062

[7]

María Anguiano, Alain Haraux. The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors. Evolution Equations & Control Theory, 2017, 6 (3) : 345-356. doi: 10.3934/eect.2017018

[8]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[9]

Anna Lenzhen, Babak Modami, Kasra Rafi. Teichmüller geodesics with $ d$-dimensional limit sets. Journal of Modern Dynamics, 2018, 12: 261-283. doi: 10.3934/jmd.2018010

[10]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[11]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124

[12]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[13]

Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096

[14]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

[15]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020079

[16]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020021

[17]

Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020135

[18]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019129

[19]

Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030

[20]

Vedran Krčadinac, Renata Vlahović Kruc. Quasi-symmetric designs on $ 56 $ points. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020086

2019 Impact Factor: 1.338

Article outline

[Back to Top]