Sub-shadowing and specification through pointwise dynamics

• Das, Pramod Kumar ^[1] ; Khan, Abdul Gaffar ^[2]
  1. [1] GITAM University

     GITAM University

     India

     
  2. [2] University of Delhi

     University of Delhi

     India

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 26, Nº. 2, 2025, págs. 691-702
• Idioma: inglés
• DOI: 10.4995/agt.2025.22742
• Enlaces
  • Texto completo
• Resumen

  • In this paper, the notion of $\overline{d}$-shadowable points for continuous maps on compact metric spaces is introduced and it is proved that an equicontinuous map (not necessarily surjective) cannot have the $\overline{d}$-shadowing property. Consequently, it is shown that a minimal continuous map with the $\overline{d}$-shadowing property is chaotic in the sense of Auslander-Yorke. Then, it is proved that the existence of a pseudo-orbital specification point, the existence of an ergodic shadowable point, the pseudo-orbital specification property and the ergodic shadowing property are equivalent. Consequently, it is shown that existence of a pseudo-orbital specification point implies the specification property. Next, it is proved that the existence of a point which is both shadowable and specification point implies the pseudo-orbital specification property, the ergodic shadowing property, the shadowing property and the $\overline{d}$-shadowing property. Consequently, it is shown that the existence of a shadowable point in a topologically mixing system implies the shadowing property as well as the specification property. Finally, it is proved that a continuous map with the specification property is mean sensitive.

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