Topological Stability and Entropy for Certain Set-valued Maps

Yu ZHANG, Yu Jun ZHU

Acta Mathematica Sinica ›› 2024, Vol. 40 ›› Issue (4) : 962-984.

PDF(349 KB)
PDF(349 KB)
Acta Mathematica Sinica ›› 2024, Vol. 40 ›› Issue (4) : 962-984. DOI: 10.1007/s10114-023-1643-7
Articles

Topological Stability and Entropy for Certain Set-valued Maps

  • Yu ZHANG, Yu Jun ZHU
Author information +
History +

Abstract

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if f is a hyperbolic endomorphism then for each ε>0 there exists a C1-neighborhood U of f such that the induced set-valued map Ff,U has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood U of f such that the induced set-valued map Ff,U has the Lipschitz shadowing property; (2) when a set-valued map F is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable; (3) if f is an expanding endomorphism then for each ε>0 there exists a C1-neighborhood U of f such that h(Ff,U,ε)=h(f); (4) when F is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of F is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

Key words

Set-valued map / orbit space / hyperbolic endomorphism / perturbation / shadowing / expansiveness / topological stability / entropy

Cite this article

Download Citations
Yu ZHANG, Yu Jun ZHU. Topological Stability and Entropy for Certain Set-valued Maps. Acta Mathematica Sinica, 2024, 40(4): 962-984 https://doi.org/10.1007/s10114-023-1643-7

References

[1] Akin, E.:Simplicial dynamical systems. Mem. Amer. Math. Soc., 140(667), x+197 pp.(1999)
[2] Anosov, D. V.:On a class of invariant sets of smooth dynamical systems. In:Proc.5th Int. Conf. Nonl. Oscill., Kiev, 2, 39-45(1970)
[3] Aubin, J., Frankowska, H.:Set-valued Analysis, Birkhauser Boston, Inc., Boston, MA, 1990
[4] Aubin, J., Frankowska, H., Lasota, A.:Poincarés recurrence theorem for set-valued dynamical systems. Ann. Polon. Math., 54, 85-91(1991)
[5] Alvin, L., Kelly, J.:Topological entropy of Markov set-valued functions. Ergodic Theory Dynam. Systems, 41(2), 321-337(2021)
[6] Bowen, R.:Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., 470, Springer, Berlin, 1975
[7] Brown, A., Hertz, F. R., Wang, Z.:Smooth ergodic theory ofZd-actions. arXiv:1610.09997(2016)
[8] Cheng, W.-C., Newhouse, S.:Pre-image entropy. Ergodic Theory Dynam. Systems, 25, 1091-1113(2005)
[9] Einsiedler, M., Lind, D.:AlgebraicZd-actions on entropy rank one. Trans. Amer. Math. Soc., 356(5), 179-1831(2004)
[10] Fiebig, D., Fiebig, U., Nitecki, Z.:Entropy and preimage sets. Ergodic Theory Dynam. Systems, 23, 1785-1806(2003)
[11] Friedland, S.:Entropy of graphs, semi-groups and groups. In:Ergodic Theory ofZd-actions, Pollicott, M., Schmidt, K.(eds.), London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 319-343(1996)
[12] Geller, W., Pollicott, M.:An entropy for Z2-actions with finite entropy generators. Fund. Math., 157, 209-220(1998)
[13] Hu, H.:Some ergodic properties of commuting diffeomorphisms. Ergodic Theory Dynam. Systems, 13, 73-100(1993)
[14] Hurley, M.:On topological entropy of maps. Ergodic Theory Dynam. Systems, 15, 557-568(1995)
[15] Ingram, W., Mahavier, W.:Inverse limits of upper semi-continuous set valued functions. Houston J. Math., 32(1), 119-130(2006)
[16] Kelly, J., Tennant, T.:Topological entropy of set-valued functions. Houston J. Math., 43(1), 263-282(2017)
[17] Kennedy, J., Nall, V.:Dynamical properties of shift maps on inverse limits with a set valued function. Ergodic Theory Dynam. Systems, 38(4), 1499-1524(2018)
[18] Kifer, Y.:Ergodic Theory of Random Transformations, Birkhä, Boston, 1986
[19] Liu, P., Qian, M.:Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Math., 1606, Springer, New York, 1995
[20] Pacifico, M., Vieitez, J.:Expansiveness, Lyapunov exponents and entropy for set valued maps. arXiv:1709.05739(2017)
[21] Pilyugin, S.:Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer, Berlin, 1999
[22] Pilyugin, S., Rieger, J.:Shadowing and inverse shadowing in set-valued dynamical systems:Contractive case. Topol. Methods Nonlinear Anal., 32(1), 139-149(2008)
[23] Pilyugin, S., Rieger, J.:Shadowing and inverse shadowing in set-valued dynamical systems:Hyperbolic case. Topol. Methods Nonlinear Anal., 32(1), 151-164(2008)
[24] Qian, M., Xie, J., Zhu, S.:Smooth ergodic theory for endomorphisms, Lecture Notes in Math., 1978. Springer-Verlag, Berlin, 2009
[25] Metzger, R., Morales, C. A., Thieullen, P.:Topological stability in set-valued dynamics. Discrete Contin. Dyn. Syst. Ser. B, 22(5), 1965-1975(2017)
[26] Miller, W.:Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems. Set-Valued Anal., 3, 181-194(1995)
[27] Miller, W., Akin, E.:Invariant measures for set-valued dynamical systems. Trans. Amer. Math. Soc., 351, 1203-1225(1999)
[28] Nitecki, Z., Przytycki, F.:Preimage entropy for mappings. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9, 1815-1843(1999)
[29] Wen, L.:Differentiable Dynamical Systems, Graduate Studies in Math., 173, Amer. Math. Soc., Providence, RI, 2016
[30] Wang, X., Wu, W., Zhu, Y.:Local entropy via preimage structure. J. Differential Equations, 317, 639-684(2022)
[31] Wu, W., Zhu, Y.:On preimage entropy, folding entropy and stable entropy. Ergodic Theory Dynam. Systems, 41(4), 1217-1249(2021)
[32] Wu, W., Zhu, Y.:Entropy via preimage structure. Adv. Math., 406(11), 108483, 45 pp (2022)
[33] Zhu, Y.:Entropy formula of randomZk-actions. Trans. Amer. Math. Soc., 369(7), 4517-4544(2017)
[34] Zhu, Y., Zhang, J., He, L.:Shadowing and inverse shadowing forC1endomorphisms. Acta Math. Sin., Engl. Ser., 22(5), 1321-1328(2006)

Funding

Supported by NSFC (Grant Nos. 11771118, 12171400)
PDF(349 KB)

164

Accesses

0

Citation

Detail

Sections
Recommended

/