Consider a dynamical system which is positively expansive and satisfies the condition of specification. We further study the topological entropy of the level sets for local Poincare recurrence, i.e. the recurrence spectrum. It turns out that the spectrum is quite irrational as any level set has the same (topological) entropy as the whole system. The erratic recurrence behavior of the orbits brings chaos. For the system concerned, we show that it contains a Xiong chaotic set C which is large in the sense that the intersection of any non-empty open set with C has the same topological entropy as the whole system. The ergodic average can be regarded as a certain recurrence average. We give multifractal analysis of the generalized spectrum for ergodic average, which incorporates the information of the set of divergence points. Note that the set of divergence points for Poincare recurrence or ergodic average has measure zero with respect to any invariant measure. (A Xiong chaotic set may has measure zero with respect to some invariant measures with full support.) The above results support the point of view that small set unobservable in measure may account for the anomalous chaotic behavior of the whole system. / The thesis is on the recurrence and chaotic behavior of a dynamical system. Let the local Poincare recurrence rate at a point be defined as the exponential rate of the first return time of the orbit into its neighborhoods defined by the Bowen metric. Given any reference invariant probability measure mu, we show that the rate equals to the local entropy of mu a.e. Hence, the integration of the rate is exactly the (measure theoretic) entropy of the measure mu. / Shu, Lin. / "January 2007." / Adviser: Ka-Sing Lau. / Source: Dissertation Abstracts International, Volume: 68-08, Section: B, page: 5286. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 83-91). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_343816 |
| Date | January 2007 |
| Contributors | Shu, Lin, Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, theses |
| Format | electronic resource, microform, microfiche, 1 online resource (iv, 91 p. : ill.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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