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
• 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.