The Entropy of Co-Compact Open Covers

Abstract

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems; and (2) it is an invariant of topological conjugation, compared to Bowen's entropy, which is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system, (R, f ), defined by f (x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More generally, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result also generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces.

Co-compact entropy and its variational principle for non-compact spaces

“…It should be pointed out that in the above example, the co-compact entropy (see definition in Section 2.2) is equal to 0, as calculated in our previous paper [19]. Furthermore, we prove the equivalence between positive co-compact entropy and the existence of p-horseshoes over the real line.…”

“…Recently, Canovas and Rodriguez [10], Malziri and Molaci [11], Liu, Wang and Wei [12], Wei, Wang and Wei [13,19], and Patrão [14] proposed kinds of definitions of topological entropy on non-compact spaces.…”

“…In this paper we prove the variational principle for non-compact systems in terms of co-compact entropy, which is defined by using the perfect map and the co-compact cover. The co-compact entropy was introduced in [13], and its properties and its relation with other entropies were further investigated in our previous paper [19].…”

“…This section reviews the concept of co-compact entropy and relevant results. The proofs were provided in our previous paper [19].…”

Entropy for semigroup actions on general topological spaces

“…Following this idea, Wei et al. [37] defined the concept of co‐compact cover to generalize the AKM topological entropy for perfect maps on Hausdorff spaces. A co‐compact cover of a Hausdorff space is a special case of admissible covering.…”