An Open Cover Specification Property

Abstract

In this paper, we define and study the topological version of the specification property, called open cover specification property. We explore its connections with different types of mixing properties, such as topological mixing, total transitivity, weak mixing, and the non-wandering property. We also establish that, in the presence of positive orbit expansivity on a compact Hausdorff space, the open cover specification property implies local eventual onto behaviour. Additionally, we calculate the topological entropy of a dynamical system on a compact Hausdorff space, showing that it can be expressed in terms of periodic points when the map satisfies both the open cover specification property and the positive orbit expansivity.