Entropy formula for random $\mathbb {Z}^k$-actions
HTML articles powered by AMS MathViewer
- by Yujun Zhu PDF
- Trans. Amer. Math. Soc. 369 (2017), 4517-4544 Request permission
Abstract:
In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random $\mathbb {Z}^k$-actions which are generated by random compositions of the generators of $\mathbb {Z}^k$-actions. Applying Pesin’s theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of $C^{2}$ random $\mathbb {Z}^k$-actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random $\mathbb {Z}^k$(or $\mathbb {Z}_+^k)$-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and $C^{1}$ expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedland’s entropy for certain $C^{2}$ $\mathbb {Z}^k$-actions.References
- L. Abramov and V. Rokhlin, The entropy of a skew product of measure-preserving transformations, Amer. Math. Soc. Transl. (Ser. 2), vol. 48, Amer. Math. Soc., Providence, RI, 1966, pp. 255–265.
- Jörg Bahnmüller and Thomas Bogenschütz, A Margulis-Ruelle inequality for random dynamical systems, Arch. Math. (Basel) 64 (1995), no. 3, 246–253. MR 1314494, DOI 10.1007/BF01188575
- Jörg Bahnmüller and Pei-Dong Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, J. Dynam. Differential Equations 10 (1998), no. 3, 425–448. MR 1646606, DOI 10.1023/A:1022653229891
- Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. MR 274707, DOI 10.1090/S0002-9947-1971-0274707-X
- Robert Burton, Karma Dajani, and Ronald Meester, Entropy for random group actions, Ergodic Theory Dynam. Systems 18 (1998), no. 1, 109–124. MR 1609487, DOI 10.1017/S0143385798097582
- Wen-Chiao Cheng and Sheldon E. Newhouse, Pre-image entropy, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1091–1113. MR 2158398, DOI 10.1017/S0143385704000240
- Manfred Einsiedler and Douglas Lind, Algebraic $\Bbb Z^d$-actions of entropy rank one, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1799–1831. MR 2031042, DOI 10.1090/S0002-9947-04-03554-8
- John Franks and MichałMisiurewicz, Topological methods in dynamics, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 547–598. MR 1928524, DOI 10.1016/S1874-575X(02)80009-1
- Shmuel Friedland, Entropy of graphs, semigroups and groups, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 319–343. MR 1411226, DOI 10.1017/CBO9780511662812.013
- W. Geller and M. Pollicott, An entropy for $\mathbf Z^2$-actions with finite entropy generators, Fund. Math. 157 (1998), no. 2-3, 209–220. Dedicated to the memory of Wiesław Szlenk. MR 1636888, DOI 10.4064/fm-157-2-3-209-220
- Hu Yi Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems 13 (1993), no. 1, 73–100. MR 1213080, DOI 10.1017/S0143385700007215
- Mike Hurley, On topological entropy of maps, Ergodic Theory Dynam. Systems 15 (1995), no. 3, 557–568. MR 1336706, DOI 10.1017/S014338570000852X
- Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
- Yuri Kifer, Ergodic theory of random transformations, Progress in Probability and Statistics, vol. 10, Birkhäuser Boston, Inc., Boston, MA, 1986. MR 884892, DOI 10.1007/978-1-4684-9175-3
- Yuri Kifer, On the topological pressure for random bundle transformations, Topology, ergodic theory, real algebraic geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, 2001, pp. 197–214. MR 1819189, DOI 10.1090/trans2/202/14
- Yuri Kifer and Pei-Dong Liu, Random dynamics, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 379–499. MR 2186245, DOI 10.1016/S1874-575X(06)80030-5
- Sergiĭ Kolyada and Ľubomír Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205–233. MR 1402417
- François Ledrappier and Jean-Marie Strelcyn, A proof of the estimation from below in Pesin’s entropy formula, Ergodic Theory Dynam. Systems 2 (1982), no. 2, 203–219 (1983). MR 693976, DOI 10.1017/S0143385700001528
- F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, DOI 10.2307/1971328
- Pei-Dong Liu and Min Qian, Smooth ergodic theory of random dynamical systems, Lecture Notes in Mathematics, vol. 1606, Springer-Verlag, Berlin, 1995. MR 1369243, DOI 10.1007/BFb0094308
- Pei-Dong Liu, Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1279–1319. MR 1855833, DOI 10.1017/S0143385701001614
- Ricardo Mañé, A proof of Pesin’s formula, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 95–102. MR 627789, DOI 10.1017/s0143385700001188
- Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
- Zbigniew Nitecki and Feliks Przytycki, Preimage entropy for mappings, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1815–1843. Discrete dynamical systems. MR 1728741, DOI 10.1142/S0218127499001309
- V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
- Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
- David Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), no. 1, 83–87. MR 516310, DOI 10.1007/BF02584795
- David Ruelle, Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 187 (1973), 237–251. MR 417391, DOI 10.1090/S0002-9947-1973-0417391-6
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
- Jin-lian Zhang and Lan-xin Chen, Lower bounds of the topological entropy for nonautonomous dynamical systems, Appl. Math. J. Chinese Univ. Ser. B 24 (2009), no. 1, 76–82. MR 2486498, DOI 10.1007/s11766-009-2013-7
- Yujun Zhu, Jinlian Zhang, and Lianfa He, Topological entropy of a sequence of monotone maps on circles, J. Korean Math. Soc. 43 (2006), no. 2, 373–382. MR 2203560, DOI 10.4134/JKMS.2006.43.2.373
- Yujun Zhu, Preimage entropy for random dynamical systems, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 829–851. MR 2318271, DOI 10.3934/dcds.2007.18.829
- Yujun Zhu, On local entropy of random transformations, Stoch. Dyn. 8 (2008), no. 2, 197–207. MR 2429200, DOI 10.1142/S0219493708002275
- Y. Zhu, W. Zhang, and E. Shi, A formula of Friedland’s entropy for $\mathbb {Z}_+^k$-actions on tori (in Chinese), Sci. China: Mathematics 44(6) (2014), 701–709.
Additional Information
- Yujun Zhu
- Affiliation: College of Mathematics and Information Science and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, Hebei, 050024, People’s Republic of China
- Received by editor(s): December 17, 2014
- Received by editor(s) in revised form: June 11, 2015
- Published electronically: December 22, 2016
- Additional Notes: The author was supported by NSFC (No. 11371120), NSFHB (No. A2014205154), BRHB (No. BR2-219) and GCCHB (No. GCC2014052)
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4517-4544
- MSC (2010): Primary 37A35, 37C85, 37H99
- DOI: https://doi.org/10.1090/tran/6798
- MathSciNet review: 3632542