Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TOPOLOGICAL ENTROPY OF SWITCHED SYSTEMS

  • Huang, Yu (School of Mathematics Sun Yat-sen University) ;
  • Zhong, Xingfu (School of Mathematics Sun Yat-sen University)
  • Received : 2017.09.26
  • Accepted : 2018.01.16
  • Published : 2018.09.01
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Abstract

For a switched system with constraint on switching sequences, which is also called a subshift action, on a metric space not necessarily compact, two kinds of topological entropies, average topological entropy and maximal topological entropy, are introduced. Then we give some properties of those topological entropies and estimate the bounds of them for some special systems, such as subshift actions generated by finite smooth maps on p-dimensional Riemannian manifold and by a family of surjective endomorphisms on a compact metrizable group. In particular, for linear switched systems on ${\mathbb{R}}^p$, we obtain a better upper bound, by joint spectral radius, which is sharper than that by Wang et al. in [42,43].

Keywords

Acknowledgement

Supported by : National Nature Science Funds of China, Sun Yat-Sen University, ordinary university of Guangdong Province

References

  1. R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. https://doi.org/10.1090/S0002-9947-1965-0175106-9
  2. L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
  3. N. E. Barabanov, On Lyapunov indicators of discrete inclusions I, Automat. Remote Control 49 (1988), no. 2, 152-157; translated from Avtomat. i Telemekh. 1988 (1988), no. 2, 40-46.
  4. M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl. 166 (1992), 21-27. https://doi.org/10.1016/0024-3795(92)90267-E
  5. A. Bis, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst. 11 (2004), no. 2-3, 639-648. https://doi.org/10.3934/dcds.2004.11.639
  6. T. Bogenschutz, Entropy, pressure, and a variational principle for random dynamical systems, Random & Computational Dynamics 1 (1992), 99-116.
  7. R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. https://doi.org/10.1090/S0002-9947-1971-0274707-X
  8. A. Bufetov, Topological entropy of free semigroup actions and skew-product transformations, J. Dynam. Control Systems 5 (1999), no. 1, 137-143. https://doi.org/10.1023/A:1021796818247
  9. J. S. Canovas, On entropy of non-autonomous discrete systems, in Progress and challenges in dynamical systems, 143-159, Springer Proc. Math. Stat., 54, Springer, Heidelberg, 2013.
  10. O. L. V. Costa, M. D. Fragoso, and R. P. Marques, Discrete-time markov jump linear systems, Automatic Control IEEE Transactions 51 (2006), no. 5, 916-917. https://doi.org/10.1109/TAC.2006.874981
  11. X. Dai, A Gel'fand-type spectral-radius formula and stability of linear constrained switching systems, Linear Algebra Appl. 436 (2012), no. 5, 1099-1113. https://doi.org/10.1016/j.laa.2011.07.029
  12. X. Dai, Some criteria for spectral finiteness of a finite subset of the real matrix space ${\mathbb{R}}^{d{\times}d}$, Linear Algebra Appl. 438 (2013), no. 6, 2717-2727. https://doi.org/10.1016/j.laa.2012.09.026
  13. X. Dai, Y. Huang, J. Liu, and M. Xiao, The finite-step realizability of the joint spectral radius of a pair of $d{\times}d$ matrices one of which being rank-one, Linear Algebra Appl. 437 (2012), no. 7, 1548-1561. https://doi.org/10.1016/j.laa.2012.04.053
  14. I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl. 161 (1992), 227-263. https://doi.org/10.1016/0024-3795(92)90012-Y
  15. E. I. Dinaburg, A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970), 19-22.
  16. A. H. Dooley and G. Zhang, Local entropy theory of a random dynamical system, Mem. Amer. Math. Soc. 233 (2015), no. 1099, vi+106 pp.
  17. L. Elsner, The generalized spectral-radius theorem: an analytic-geometric proof, Linear Algebra Appl. 220 (1995), 151-159. https://doi.org/10.1016/0024-3795(93)00320-Y
  18. S. Friedland, Entropy of graphs, semigroups and groups, in Ergodic theory of $Z^d$ actions (Warwick, 1993-1994), 319-343, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996.
  19. G. Froyland and O. Stancevic, Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems, Stoch. Dyn. 13 (2013), no. 4, 1350004, 26 pp. https://doi.org/10.1142/S0219493713500044
  20. N. Guglielmi and V. Protasov, Exact computation of joint spectral characteristics of linear operators, Found. Comput. Math. 13 (2013), no. 1, 37-97. https://doi.org/10.1007/s10208-012-9121-0
  21. K. H. Hofmann and L. N. Stojanov, Topological entropy of group and semigroup actions, Adv. Math. 115 (1995), no. 1, 54-98. https://doi.org/10.1006/aima.1995.1050
  22. W. Huang, Entropy, chaos and weak horseshoe for infinite dimensional random dynamical systems, Mathematics, 2015.
  23. C. Kawan, Metric entropy of nonautonomous dynamical systems, Nonauton. Dyn. Syst. 1 (2014), 26-52.
  24. C. Kawan and Y. Latushkin, Some results on the entropy of non-autonomous dynamical systems, Dyn. Syst. 31 (2016), no. 3, 251-279. https://doi.org/10.1080/14689367.2015.1111299
  25. Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhauser Boston, Inc., Boston, MA, 1986.
  26. A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861-864.
  27. S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205-233.
  28. V. Kozyakin, Matrix products with constraints on the sliding block relative frequencies of different factors, Linear Algebra Appl. 457 (2014), 244-260. https://doi.org/10.1016/j.laa.2014.05.016
  29. V. Kozyakin, Constructive stability and stabilizability of positive linear discrete-time switching systems, arXiv:1511.05665v1, 2015.
  30. P. Kurka, Topological and Symbolic Dynamics, Cours Specialises, 11, Societe Mathematique de France, Paris, 2003.
  31. R. Langevin and F. Przytycki, Entropie de l'image inverse d'une application, Bull. Soc. Math. France 120 (1992), no. 2, 237-250. https://doi.org/10.24033/bsmf.2185
  32. R. Langevin and P. Walczak, Entropie d'une dynamique, C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), no. 1, 141-144.
  33. E. Mihailescu and M. Urbanski, Inverse topological pressure with applications to holomorphic dynamics of several complex variables, Commun. Contemp. Math. 6 (2004), no. 4, 653-679. https://doi.org/10.1142/S0219199704001446
  34. E. Mihailescu and M. Urbanski, Inverse pressure estimates and the independence of stable dimension for non-invertible maps, Canad. J. Math. 60 (2008), no. 3, 658-684. https://doi.org/10.4153/CJM-2008-029-2
  35. E. Mihailescu and M. Urbanski, Random countable iterated function systems with overlaps and applications, Adv. Math. 298 (2016), 726-758. https://doi.org/10.1016/j.aim.2016.05.002
  36. Z. H. Nitecki, Topological entropy and the preimage structure of maps, Real Anal. Exchange 29 (2003/04), no. 1, 9-41. https://doi.org/10.14321/realanalexch.29.1.0009
  37. G.-C. Rota and G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 379-381.
  38. M.-H. Shih, J.-W. Wu, and C.-T. Pang, Asymptotic stability and generalized Gelfand spectral radius formula, Linear Algebra Appl. 252 (1997), 61-70. https://doi.org/10.1016/0024-3795(95)00592-7
  39. J. Sinai, On the concept of entropy for a dynamic system, Dokl. Akad. Nauk SSSR 124 (1959), 768-771.
  40. Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design. Springer-Verlag, 2006.
  41. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.
  42. Y. Wang and D. Ma, On the topological entropy of a semigroup of continuous maps, J. Math. Anal. Appl. 427 (2015), no. 2, 1084-1100. https://doi.org/10.1016/j.jmaa.2015.02.082
  43. Y. Wang, D. Ma, and X. Lin, On the topological entropy of free semigroup actions, J. Math. Anal. Appl. 435 (2016), no. 2, 1573-1590. https://doi.org/10.1016/j.jmaa.2015.11.038
  44. Y. Zhu, Z. Liu, X. Xu, and W. Zhang, Entropy of nonautonomous dynamical systems, J. Korean Math. Soc. 49 (2012), no. 1, 165-185. https://doi.org/10.4134/JKMS.2012.49.1.165