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

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.

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 and 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.

[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. 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.

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

[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.

[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. 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.

[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. 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.

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

[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.

[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. 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.

[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.

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