Topological Entropy and Sequence Entropy for Hom Tree-Shifts on Unexpandable Trees

• Jung-Chao Ban ^[1]
  1. [1] National Chengchi University & National Taiwan University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 3, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This article explores the topological entropy and topological sequence entropy of hom tree-shifts on unexpandable trees. Regarding topological entropy, we establish that the entropy, denoted as on an unexpandable tree, equals the entropy h(X) of the base shift X when X is a subshift satisfying the almost specification property. Additionally, we derive some fundamental properties such as entropy approximation and the denseness of entropy for subsystems. Concerning topological sequence entropy, we show that the set of sequence entropies of hom tree-shifts with a base shift is generated by an irreducible matrix A, forming a subset of . Precisely, these entropies correspond to the logarithms of the largest cardinalities of the periodic classes of A.

• Referencias bibliográficas
  • Aubrun, N., Beal, M.-P.: Tree-shifts of finite type. Theoret. Comput. Sci. 459, 16–25 (2012)
  • Aubrun, N., Béal, M.-P.: Sofic tree-shifts. Theory Comput. Syst. 53(4), 621–644 (2013)
  • Balibrea, F., López, V., Pena, J.: Some results on entropy and sequence entropy. Int. J. Bifurc. Chaos 9(09), 1731–1742 (1999)
  • Ban, J.-C., Chang, C.-H.: Tree-shifts: irreducibility, mixing, and chaos of tree-shifts. Trans. Am. Math. Soc. 369, 8389–8407 (2017)
  • Ban, J.-C., Chang, C.-H.: Tree-shifts: the entropy of tree-shifts of finite type. Nonlinearity 30, 2785–2804 (2017)
  • Ban, J.-C., Chang, C.-H., Hu, W.-G., Lai, G.-Y., Wu, Y.-L.: An Analogue of Topological Sequence Entropy for Markov Hom Tree-Shifts, Studia...
  • Ban, J.-C., Chang, C.-H., Hu, W.-G., Wu, Y.-L.: On structure of topological entropy for tree-shift of finite type. J. Differ. Equ. 292, 325–353...
  • Ban, J.-C., Chang, C.-H., Hu, W.-G., Wu, Y.-L.: Topological entropy for shifts of finite type over z and trees. Theoret. Comput. Sci. 930,...
  • Ban, J.-C., Chang, C.-H., Wu, Y.-L., Wu, Y.-Y.: Stem and topological entropy on cayley trees. Math. Phys. Anal. Geom. 25(1), 1–40 (2022)
  • Ban, J.-C., Hu, W.-G., Lai, G.-Y.: The entropy structures of axial products on and trees, arXiv preprint arXiv:2303.13011 (2023)
  • Baxter, R.J., Enting, I.G., Tsang, S.K.: Hard-square lattice gas. J. Stat. Phys. 22, 465–489 (1980)
  • Bland, R., McGoff, K., Pavlov, R.: Subsystem Entropies of Shifts of Finite Type and Sofic Shifts on Countable Amenable Groups, Ergodic Theory...
  • Cánovas, J. S.: A Guide to Topological Sequence Entropy, Progress in Mathematical Biology Research, 101–139 (2008)
  • Ceccherini-Silberstein, T., Coornaert, M.: Cellular Automata and Groups. Springer Science & Business Media (2010)
  • Dembo, A., Montanari, A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2), 565–592 (2010)
  • Dembo, A., Montanari, A., Sun, N.: Factor models on locally tree-like graphs. Ann. Probab. 41(6), 4162–4213 (2013)
  • Dembo, A., Zeitouni, O.: LDP for Finite Dimensional Spaces, pp. 11–70. Springer, Large deviations techniques and applications (2009)
  • Desai, A.: Subsystem entropy for sofic shifts. Indag. Math. 17(3), 353–359 (2006)
  • Dou, D., Ye, X.-D., Zhang, G.-H.: Entropy sequences and maximal entropy sets. Nonlinearity 19(1), 53 (2005)
  • Franzová, N., Smítal, J.: Positive sequence topological entropy characterizes chaotic maps. Proc. Am. Math. Soc. 112(4), 1083–1086 (1991)
  • Georgii, H. O.: Gibbs Measures and Phase Transitions, vol. 9, Walter de Gruyter, (2011)
  • Goodman, T.: Topological sequence entropy. Proc. Lond. Math. Soc. 3(2), 331–350 (1974)
  • Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. 171, 2011–2038 (2010)
  • Hric, R.: Topological sequence entropy for maps of the circle. Comment. Math. Univ. Carol. 41(1), 53–59 (2000)
  • Huang, W., Shao, S., Ye, X.-D.: Mixing via sequence entropy. Contemp. Math. 385, 101–122 (2005)
  • Huang, W., Ye, X.-D.: Combinatorial lemmas and applications to dynamics. Adv. Math. 220(6), 1689–1716 (2009)
  • Kamae, T., Zamboni, L.: Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Theory Dynam. Syst. 22(4), 1191–1199...
  • Kuang, R., Yang, Y.-N.: Supremum topological sequence entropy of circle maps. Topol. Appl. 295, 107670 (2021)
  • Kushnirenko, A.G.: On metric invariants of entropy type. Russ. Math. Surv. 22(5), 53 (1967)
  • Lieb, E.H.: Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18(17), 692 (1967)
  • Lind, D.: The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Theory Dynam. Syst. 4(2), 283–300 (1984)
  • Lind, D.: Perturbations of shifts of finite type. SIAM J. Discret. Math. 2(3), 350–365 (1989)
  • Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)
  • Pavlov, R.: Perturbations of multidimensional shifts of finite type. Ergod. Theory Dynam. Syst. 31(2), 483–526 (2011)
  • Petersen, K., Salama, I.: Tree shift topological entropy. Theoret. Comput. Sci. 743, 64–71 (2018)
  • Petersen, K., Salama, I.: Entropy on regular trees. Discret. Contin. Dynam. Syst. 40(7), 4453 (2020)
  • Quas, A., Trow, P.: Subshifts of multi-dimensional shifts of finite type. Ergod. Theory Dynam. Syst. 20(3), 859–874 (2000)
  • Saleski, A.: Sequence entropy and mixing. J. Math. Anal. Appl. 60(1), 58–66 (1977)
  • Spitzer, F.: Markov random fields on an infinite tree. Ann. Probab. 3(3), 387–398 (1975)
  • Tan, F.: The set of sequence entropies for graph maps. Topol. Appl. 158(3), 533–541 (2011)
  • Tan, F., Ye, X.D., Zhang, R.-F.: The set of sequence entropies for a given space. Nonlinearity 23(1), 159 (2009)