Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric

• J. Muentes ^[1] ; A. J. Becker ^[2] ; A.T. Baraviera ^[3] ; É. Scopel ^[4]
  1. [1] Universidad Tecnológica de Bolívar

     Universidad Tecnológica de Bolívar

     Colombia

     
  2. [2] Universidade Federal de Santa Maria

     Universidade Federal de Santa Maria

     Brasil

     
  3. [3] Universidade Federal do Rio Grande do Sul

     Universidade Federal do Rio Grande do Sul

     Brasil

     
  4. [4] Instituto Federal de Educação, Ciência e Tecnologia do Rio Grande do Sul

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01100-1
• Enlaces
  • Texto completo
• Resumen

  • Let f : M → M be a continuous map on a compact metric space M equipped with a fixed metric d, and let τ be the topology on M induced by d. We denote by M(τ ) the set consisting of all metrics on M that are equivalent to d. Let mdimM(M, d, f ) and mdimH(M, d, f ) be, respectively, the metric mean dimension and mean Hausdorff dimension of f. First, we will establish some fundamental properties of the mean Hausdorff dimension. Furthermore, it is important to note that mdimM(M, d, f ) and mdimH(M, d, f ) depend on the metric d chosen for M. In this work, we will prove that, for a fixed dynamical system f : M → M, the functions mdimM(M, f ) :

    M(τ ) → R ∪ {∞} and mdimH(M, f ) : M(τ ) → R ∪ {∞} are not continuous, where mdimM(M, f )(ρ) = mdimM(M,ρ, f ) and mdimH(M, f )(ρ) = mdimH(M,ρ, f ) for any ρ ∈ M(τ ). Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.

• Referencias bibliográficas
  • 1. Acevedo, J.M.: Genericity of continuous maps with positive metric mean dimension. RM 77(1), 1–30 (2022)
  • 2. Acevedo, J.M.: Genericity of homeomorphisms with full mean Hausdorff dimension. Regul. Chaotic Dyn. 29, 1–17 (2024)
  • 3. Acevedo, J.M., Romaña, S., Arias, R.: Density of the level sets of the metric mean dimension for homeomorphisms. J. Dyn. Differ. Equ. 1–14...
  • 4. Backes, L., Rodrigues, F.B.: A Variational Principle for the Metric Mean Dimension of Level Sets. IEEE Trans. Inf. Theory 69(9), 5485–5496...
  • 5. Carvalho, M., Pessil, G., Varandas, P.: A convex analysis approach to the metric mean dimension: limits of scaled pressures and variational...
  • 6. Carvalho, M., Rodrigues, F.B., Varandas, P.: Generic homeomorphisms have full metric mean dimension. Ergod. Theory Dyn. Syst. 42(1), 40–64...
  • 7. Cheng, D., Li, Z., Selmi, B.: Upper metric mean dimensions with potential on subsets. Nonlinearity 34(2), 852 (2021)
  • 8. Dou, D.: Minimal subshifts of arbitrary mean topological dimension. Discrete Contin. Dyn. Syst. 37(3), 1411–1424 (2016)
  • 9. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley (2004)
  • 10. Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49...
  • 11. Gromov, M.: Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2(4), 323–415 (1999)
  • 12. Gutman, Y.: Embedding topological dynamical systems with periodic points in cubical shifts. Ergod. Theory Dyn. Syst. 37(2), 512–538 (2017)
  • 13. Gutman, Y., Qiao, Y., Tsukamoto, M.: Application of signal analysis to the embedding problem of Zk -actions. Geom. Funct. Anal. 29(5),...
  • 14. Kloeckner, B., Mihalache, N.: An invitation to rough dynamics: zipper maps. arXiv preprint arXiv:2210.13038 (2022)
  • 15. Lacerda, G., Romaña, S.: Typical conservative homeomorphisms have total metric mean dimension. arXiv preprint arXiv:2311.03607 (2023)
  • 16. Lindenstrauss, E., Weiss, B.: Mean topological dimension. Israel J. Math. 115(1), 1–24 (2000)
  • 17. Liu, Y., Selmi, B., Li, Z.: On the mean fractal dimensions of the Cartesian product sets. Chaos, Solitons & Fractals 180, 114503 (2024)
  • 18. Lindenstrauss, E., Tsukamoto, M.: Double variational principle for mean dimension. Geom. Funct. Anal. 29(4), 1048–1109 (2019)
  • 19. Lindenstrauss, E.: Mean dimension, small entropy factors and an embedding theorem. Inst. Hautes Etudes Sci. Publ. Math. 89, 227–262 (1999)
  • 20. Ma, X., Yang, J., Chen, E.: Mean topological dimension for random bundle transformations. Ergod. Theory Dyn. Syst. 39(4), 1020–1041 (2019)
  • 21. Shinoda, M., Tsukamoto, M.: Symbolic dynamics in mean dimension theory. Ergod. Theory Dyn. Syst. 41(8), 2542–2560 (2021)
  • 22. Salat, T., Toth, J., Zsilinszky, L.: Metric space of metrics defined on a given set. Real Anal. Exch. 18(1), 225–231 (1992-1993)
  • 23. Tsukamoto, M.: Mean dimension of full shifts. Israel J. Math. 230(1), 183–193 (2019)
  • 24. Velozo, A., Velozo, R.: Rate distortion theory, metric mean dimension and measure theoretic entropy. arXiv preprint arXiv:1707.05762 (2017)
  • 25. Yang, R., Chen, E., Zhou, X.: Bowen’s equations for upper metric mean dimension with potential. Nonlinearity 35(9), 4905 (2022)