On the Entropy Points and Shadowing in Uniform Spaces

Abstract

We introduce and study the topological concepts of entropy points, expansivity and shadowing property for dynamical systems on noncompact nonmetrizable spaces, which generalize the relevant concepts for metric spaces. We also obtain various properties on uniform entropy on noncompact nonmetrizable spaces. The main result is a theorem which yields a relation between topological shadowing property and positive uniform entropy.

Recurrent sets for endomorphisms of topological groups

“…DOI: 10.24193/mathcluj.2024. 1.02 results about equicontinuity to the uniform spaces; Das et al [13] generalized spectral decomposition theorem to the uniform spaces; The authors of [3] generalized concepts of entropy points, expansivity and the shadowing property for dynamical systems on uniform spaces and obtained a relation between the topological shadowing property and the positive uniform entropy. For more results on properties of dynamical systems in purely topological terms, one is referred to [2,14,22,[26][27][28].…”

Topological chain and shadowing properties of dynamical systems on uniform spaces

“…Then, we [15] proved that a point transitive dynamical system is either sensitive or almost equicontinuous. Recently, we [3,4] generalized concepts of entropy points, expansivity and shadowing property for dynamical systems to uniform spaces and obtained a relation between topological shadowing property and positive uniform entropy. Shah et al [13] showed that a dynamical system on a totally bounded uniform space which is topologically shadowing, mixing, and topologically expansive has the topological specification property.…”

On periodic shadowing, transitivity, chain mixing and expansivity in uniform dynamical systems

“…The notion of positively expansive maps were introduced in uniform spaces [33,16]. There are more results of dynamical systems on uniform spaces in [3,36].…”