Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces
This paper is concerned with some stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces. A new form of sensitivity is introduced, called ergodic sensitivity. It is shown that, on a metric probability space with a fully supported measure, if a measure-preservin...
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| Format: | Article |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/769523 |
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| author | Risong Li Yuming Shi |
| author_facet | Risong Li Yuming Shi |
| author_sort | Risong Li |
| collection | DOAJ |
| description | This paper is concerned with some stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces. A new form of sensitivity is introduced, called ergodic sensitivity. It is shown that, on a metric probability space with a fully supported measure, if a measure-preserving map is weak mixing, then it is ergodically sensitive and multisensitive; and if it is strong mixing, then it is cofinitely sensitive, where it is not required that the map is continuous and the space is compact. Similar results for measure-preserving semiflows are obtained, where it is required in a result about ergodic sensitivity that the space is compact in some sense and the semiflow is continuous. In addition, relationships between some sensitive properties of a map and its iterations are discussed, including syndetic sensitivity, cofinite sensitivity, ergodic sensitivity as well as usual sensitivity, n-sensitivity, and multisensitivity. Moreover, it is shown that multisensitivity, cofinite sensitivity, and ergodic sensitivity can be lifted up by a semiopen factor map. |
| format | Article |
| id | doaj-art-c4b09a66db704ad7908153a8fcdca91d |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-c4b09a66db704ad7908153a8fcdca91d2025-08-20T02:18:47ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/769523769523Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability SpacesRisong Li0Yuming Shi1School of Science, Guangdong Ocean University, Zhanjiang 524025, ChinaDepartment of Mathematics, Shandong University, Jinan, Shandong 250100, ChinaThis paper is concerned with some stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces. A new form of sensitivity is introduced, called ergodic sensitivity. It is shown that, on a metric probability space with a fully supported measure, if a measure-preserving map is weak mixing, then it is ergodically sensitive and multisensitive; and if it is strong mixing, then it is cofinitely sensitive, where it is not required that the map is continuous and the space is compact. Similar results for measure-preserving semiflows are obtained, where it is required in a result about ergodic sensitivity that the space is compact in some sense and the semiflow is continuous. In addition, relationships between some sensitive properties of a map and its iterations are discussed, including syndetic sensitivity, cofinite sensitivity, ergodic sensitivity as well as usual sensitivity, n-sensitivity, and multisensitivity. Moreover, it is shown that multisensitivity, cofinite sensitivity, and ergodic sensitivity can be lifted up by a semiopen factor map.http://dx.doi.org/10.1155/2014/769523 |
| spellingShingle | Risong Li Yuming Shi Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces Abstract and Applied Analysis |
| title | Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces |
| title_full | Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces |
| title_fullStr | Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces |
| title_full_unstemmed | Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces |
| title_short | Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces |
| title_sort | stronger forms of sensitivity for measure preserving maps and semiflows on probability spaces |
| url | http://dx.doi.org/10.1155/2014/769523 |
| work_keys_str_mv | AT risongli strongerformsofsensitivityformeasurepreservingmapsandsemiflowsonprobabilityspaces AT yumingshi strongerformsofsensitivityformeasurepreservingmapsandsemiflowsonprobabilityspaces |