半群作用之熵結構與維度不變量關係

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Title:	半群作用之熵結構與維度不變量關係 Entropy for Semigroup Actions and Relation between Dimension Invariants
Authors:	鄭文巧
Contributors:	應用數學系
Date:	2015-08
Issue Date:	2015-09-04 11:15:01 (UTC+8)
Abstract:	預計將研讀entropy和dimension進一步論文，更深入研究具有半群隨機動力系統的軌跡函數之 entropy架構.然後思考在拓樸群空間下如何描述條件熵（conditional entropy)，拓樸熵(topological entropy)與郝思多維度（Hausdorff dimension)，盒子維度（box dimension)之關係.為了將傳統碎形幾何方法延伸到正向對映隨機動力系統的實務應用之中，本計劃將詳細研讀維度方法基本分析，預測軌跡系統維度之基本性質.我們舉了很多半群隨機動力系統的應用實例，期望藉以mass distribution principle分析過程，研究不變集合dimension之測度方法與應用.在這研究中，我們並比較此一系統分析模式在測量碎形幾何過程與傳統測度的優異性.更期望預估半群隨機動力系統之條件熵將有上半連續（upper semicontinuity)的性質，共同擁有遍歷架構.當系統的自由度無限增大時，遍歷的可能性也就越來越增大，想理下 Shannon-McMillan-Breimann定理在動力系統中，共同擁有不變測度.將審查是否有相似成果，因為混*;屯系統的發展正處於實質性應用開發的研究階段，符號動力學（symbolic dynamics)已是非線性半群隨機系統研究的核心部分.在不變系統架構的假設之下，預估把多函數半群動力系統的探討，提起到符號動力學上研究它的遞移性（transitivity)、遞迴性（recurrence)及周期性質.最後，計劃要把碎形與條件熵具體的結果應用在電腦晨，例如壓縮、放大電腦圖片. Plan to study entropy and dimension advanced paper and research those entropy invariants under semigroup actions. Then, investigate the relationships among conditional entropy, topological entropy, Hausdorff dimension and box dimension. This project will use the approximation of fractal geometry to analysis properties of dimension and predict the structure of orbits. We hope to obtain some properties of invariant sets by using mass distribution principle. Symbolic space for semigroup action is also discussed, including transitivity, recurrence and periodicity. Finally, we use those results into computer graphing.
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