The Entropy of Co-Compact Open Covers

• Published in: Entropy (Basel, Switzerland) Vol. 15; no. 7; pp. 2464 - 2479
• Main Authors: Wei, Zheng, Wang, Yangeng, Wei, Guo, Wang, Tonghui, Bourquin, Steven
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.07.2013
• Subjects: co-compact open cover; Lebesgue number; perfect mapping; topological conjugation; topological dynamical system; topological entropy
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e15072464

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.