The Entropy of Co-Compact Open Covers
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 ent...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2013-06-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/15/7/2464 |
| _version_ | 1817989785891897344 |
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| author | Steven Bourquin Tonghui Wang Guo Wei Yangeng Wang Zheng Wei |
| author_facet | Steven Bourquin Tonghui Wang Guo Wei Yangeng Wang Zheng Wei |
| author_sort | Steven Bourquin |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T00:50:26Z |
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| id | doaj.art-15e7ebd10f0949f89e62c660a3fa01c2 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T00:50:26Z |
| publishDate | 2013-06-01 |
| publisher | MDPI AG |
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| series | Entropy |
| spelling | doaj.art-15e7ebd10f0949f89e62c660a3fa01c22022-12-22T02:21:48ZengMDPI AGEntropy1099-43002013-06-011572464247910.3390/e15072464The Entropy of Co-Compact Open CoversSteven BourquinTonghui WangGuo WeiYangeng WangZheng WeiCo-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.http://www.mdpi.com/1099-4300/15/7/2464topological dynamical systemperfect mappingco-compact open covertopological entropytopological conjugationLebesgue number |
| spellingShingle | Steven Bourquin Tonghui Wang Guo Wei Yangeng Wang Zheng Wei The Entropy of Co-Compact Open Covers Entropy topological dynamical system perfect mapping co-compact open cover topological entropy topological conjugation Lebesgue number |
| title | The Entropy of Co-Compact Open Covers |
| title_full | The Entropy of Co-Compact Open Covers |
| title_fullStr | The Entropy of Co-Compact Open Covers |
| title_full_unstemmed | The Entropy of Co-Compact Open Covers |
| title_short | The Entropy of Co-Compact Open Covers |
| title_sort | entropy of co compact open covers |
| topic | topological dynamical system perfect mapping co-compact open cover topological entropy topological conjugation Lebesgue number |
| url | http://www.mdpi.com/1099-4300/15/7/2464 |
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