The Relationships Between Discrete Dynamical Systems in Topological Spaces and their respective Hyperextensions to sets of Compact Spaces

Abstract

Several studies have been carried out related to the analysis of the relationship with respect to the dynamic properties of f and its hyperextension ̄f. However, the literature regarding the analysis of the effects of individual and collective chaos on their behaviour is scarce. Therefore, in this article several conjectures and questions are established according to the affectation of individual chaos in an ecosystem and its chaotic behaviour within the dynamics of this ecosystem, but as a whole. Thus, in the first instance, an introduction to the conceptualization of topological transitivity, chaos in the Devaney sense and how they are specified in continuous linear operators arranged in a Fréchet space (hypercyclic operators) will be established. In addition, the different notions of chaos that can occur depending on the relationship of the function, and its hyperextension will be described, to finally corroborate the present chaos with greater force than Devaney’saccording to the strong periodic specification property, the same one that applies to both f and a ̄f with the purpose of verifying the directionality in which individual and collective chaos can occur.