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March  2021, 41(3): 1271-1296. doi: 10.3934/dcds.2020317

On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property

Magdalena Foryś-Krawiec 1, , Jiří Kupka 1, , Piotr Oprocha 2,, and  Xueting Tian 3,

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  March 2021 Early access  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.

Keywords: Shadowing, entropy dense, irregular set, $ \Phi $-limit set.
Mathematics Subject Classification: 37B20, 37B40.
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, 2021, 41 (3) : 1271-1296. 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

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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. 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.  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. Ercai, T. 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. Fan, D.-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. 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.  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. 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.  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. Ercai, T. 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. Fan, D.-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. 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.  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

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