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The Entropy of Co-Compact Open Covers
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88001, USA
Department of Mathematics, Northwest University, Xi'an, Shaanxi 710069, China
Department of Mathematics & Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA
* Author to whom correspondence should be addressed.
Received: 3 April 2013; in revised form: 8 June 2013 / Accepted: 18 June 2013 / Published: 24 June 2013
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.
Keywords: topological dynamical system; perfect mapping; co-compact open cover; topological entropy; topological conjugation; Lebesgue number
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MDPI and ACS Style
Wei, Z.; Wang, Y.; Wei, G.; Wang, T.; Bourquin, S. The Entropy of Co-Compact Open Covers. Entropy 2013, 15, 2464-2479.
Wei Z, Wang Y, Wei G, Wang T, Bourquin S. The Entropy of Co-Compact Open Covers. Entropy. 2013; 15(7):2464-2479.
Wei, Zheng; Wang, Yangeng; Wei, Guo; Wang, Tonghui; Bourquin, Steven. 2013. "The Entropy of Co-Compact Open Covers." Entropy 15, no. 7: 2464-2479.