Entropy of Axial Products on Nd and Trees

• Jung-Chao Ban ^[1] ; Wen-Guei Hu ^[2] ; Guan-Yu Lai ^[3]
  1. [1] National Taiwan University

     National Taiwan University

     Taiwán

     
  2. [2] Sichuan University

     Sichuan University

     China

     
  3. [3] National Chengchi University

     National Chengchi University

     Taiwán

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
• Idioma: inglés
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• Resumen

  • In this paper, we first concentrate on the possible values and dense property of entropies for isotropic and anisotropic axial products of subshifts of finite type (SFTs) on Nd and d-tree Td. We prove that the entropies of isotropic and anisotropic axial products of SFTs on Nd are dense in [0, -], and the same result also holds for anisotropic axial products of SFTs on Td. However, the result is no longer true for isotropic axial products of SFTs on Td. Next, motivated by the work of Johnson et al. (Complex Syst 17(3):243, 2007), and Schraudner (Discrete Contin Dyn Syst 26(1):333, 2010), we establish the formulae and structures for entropies of full axial extension shifts on Nd and Td. Combining the aforementioned results with the findings on the surface entropy for multiplicative integer systems (Ban et al. in J Math Phys 64:16, 2023) on Nd enables us to estimate the surface entropy for the full axial extension shifts on Td. Finally, we extend the results of full axial extension shifts on Td to general trees.

• Referencias bibliográficas
  • 1. Assani, I., Presser, K.: A survey of the return times theorem, ergodic theory and dynamical systems. In: Proceedings of the Ergodic Theory...
  • 2. Aubrun, N., Beal, M.-P.: Tree-shifts of finite type. Theor. Comput. Sci. 459, 16–25 (2012)
  • 3. Ban, J.-C., Chang, C.-H.: Tree-shifts: irreducibility, mixing, and chaos of tree-shifts. Trans. Am. Math. Soc. 369, 8389–8407 (2017)
  • 4. Ban, J.-C., Chang, C.-H.: Tree-shifts: the entropy of tree-shifts of finite type. Nonlinearity 30, 2785– 2804 (2017)
  • 5. 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,...
  • 6. Ban, J.-C., Chang, C.-H., Hu, W.-G., Wu, Y.-L.: Topological entropy for shifts of finite type over Z and trees. Theor. Comput. Sci. 930,...
  • 7. 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 (2022)
  • 8. Ban, J.-C., Hu, W.-G., Lai, G.-Y.: Boundary complexity and surface entropy of 2-multiplicative integer systems on Nd . J. Math. Phys. 64,...
  • 9. Callard, A., Vanier, P.: Computational characterization of surface entropies for Z2 subshifts of finite type. In: Leibniz International...
  • 10. Ceccherini-Silberstein, T., Coornaert, M.: Cellular Automata and Groups. Springer, Berlin (2010)
  • 11. Chandgotia, N., Marcus, B.: Mixing properties for hom-shifts and the distance between walks on associated graphs. Pac. J. Math. 294, 41–69...
  • 12. Desai, A.: Subsystem entropy for Zd sofic shifts. Indag. Math. 17(3), 353–359 (2006)
  • 13. Fan, A.H.: Some aspects of multifractal analysis. In: Geometry and Analysis of Fractals, pp. 115–145. Springer, Berlin (2014)
  • 14. Fan, A.-H., Liao, L., Ma, J.-H.: Level sets of multiple ergodic averages. Monatshefte für Mathematik 168(1), 17–26 (2012)
  • 15. Hochman, M., Meyerovitch, T.: A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. 171, 2011–2038...
  • 16. Johnson, A., Kass, S., Madden, K.: Projectional entropy in higher dimensional shifts of finite type. Complex Syst. 17(3), 243 (2007)
  • 17. Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Theory Dyn. Syst....
  • 18. Kolyada, S., Snoha, L.: Some aspects of topological transitivity-a survey. Grazer Math. Ber. 334, 3–35 (1997)
  • 19. Lind, D.: The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Theory Dyn. Syst. 4(2), 283–300...
  • 20. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)
  • 21. Louidor, E., Marcus, B., Pavlov, R.: Independence entropy of Zd shift spaces. Acta Appl. Math. 126(1), 297–317 (2013)
  • 22. Meyerovitch, T.: Growth-type invariants for Zd subshifts of finite type and classes arithmetical of real numbers. Invent. Math. 184(3),...
  • 23. Meyerovitch, T., Pavlov, R.: On independence and entropy for high-dimensional isotropic subshifts. Proc. Lond. Math. Soc. 109(4), 921–945...
  • 24. Pace, D.: Surface entropy of shifts of finite type. Ph.D. thesis, University of Denver (2018)
  • 25. Petersen, K., Salama, I.: Tree shift topological entropy. Theor. Comput. Sci. 743, 64–71 (2018)
  • 26. Petersen, K., Salama, I.: Entropy on regular trees. Discrete Contin. Dyn. Syst. 40(7), 4453 (2020)
  • 27. Petersen, K., Salama, I.: Asymptotic pressure on some self-similar trees. Stoch. Dyn. (2021)
  • 28. Schraudner, M.: Projectional entropy and the electrical wire shift. Discrete Contin. Dyn. Syst. 26(1), 333 (2010)