| 摘要: | 碎形在複雜系統、圖形學、基因、資訊運用上都扮演相當重要的地位, 碎形 幾何是近幾年新興的一門數學分支, 仍是一個關於數學外型的研究. 自然界中的碎 形, 如地上的花、河流、海岸線, 天空中的星、雲、細菌的成長、血管的形狀等等, 都可以用碎形描述其基本結構. 1948 年, Shannon 採用了熵(entropy) 字眼, 把 “冗贅＂(redundancy) 概念給予量化. 測度熵於1958 年, 由俄羅斯數學家 Kolmogorov 從信息理論引入動力系統和遍歷理論中. 1965 年, 拓樸熵 (topological entropy)被定義在緊緻空間. 熵代表系統運作的「不確定性」或「亂 度」. 在多函數混沌系統中, 維度詮釋不變集合之構造, 熵可描述系統的穩定狀態. 我們運用碎形幾何及熵方法來描述、分析大自然中的複雜圖案. 以已知不變測度信 息下, 預測系統不變集合之大小與亂度為動機, 本計劃意欲集中於多個正向對映 混沌系統, 深入探討不變集合之碎形幾何架構和熵理論之基本性質. 預估混沌系統之測度熵將有中凸的性質, 共同擁有遍歷架構, 亦期望發現 Birkhoff Ergodic Theorem 相似成果. 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, 想理解 Shannon-McMillan-Breimann 定理在混沌系統中, 共 同擁有不變測度. 是否有相似成果, 將審查 power rule, product rule, affinity, generator 是否保持. 因為混沌通訊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)已是非線性混沌系統研究的核心部 分. 在不變更系統架構的假設之下, 預估把多函數混沌系統的探討, 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.
Measure-theoretic entropy and topological entropy are two of the im- portant concepts in chaotic dynamical systems. Measure-theoretic entropy describes the exponential growth rate of the statistically signi¯cant orbits. Topological entropy characterizes the total exponential complexity of the orbit structure with a single number. Fractal geometry is more and more important in information theory, graph theory and chaotic systems. Haus- dor® dimension describes the shape of invariant sets and justi¯es those basic structures. In 1958, Kolmogorov introduced those concepts into Ergodic tho- ery. Now we study the same methods and apply those concept to analysis those complexity of the graph. During this project, we will focus in multiplies chaotic systems, research those basic propositions of fractal geometry and entropy. Under the set of all invariant probability measures, a series of rules has been discussed to deter- mine the connection between measure-theoretic entropy notions and Haus- dor® dimension. Those studies prove new convex proposition and give similar theory of Birkho® Ergodic Theory. The entropy concept can be localized by restricting in the symbolic space. During this project, we investigate more advanced results of the transitivity, recurrence and periodicity. We predict this process requires adaption and modi¯cation of a number of techniques in the literature of ergodic theory and topological dynamics. |
