自由半群之次可加潛能量及碎型維度性質

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题名:	自由半群之次可加潛能量及碎型維度性質 Sub-Additive Potentials and Fractal Dimension Properties for Free Semigroup
作者:	鄭文巧
贡献者:	應用數學系
日期:	2017-2018
上传时间:	2018-03-07 12:59:51 (UTC+8)
摘要:	獲得計畫其間,個人打算與河北師範大學朱玉峻老師共同研讀論文 “Invariance entropy for topological semigroup actions” by Colonius, Fukuoka 及 Santana . 同時, 針對半群次可加潛能量條件熵參數與碎形之估計量, 我們提出適當可行估計法的評判準則,例如, generator 原創存在的工作與 local dimension 計算方法. 期望有進一步成果.亦理解共同不變測度在計量方法的演算過程中, 是否符合人類邏輯推論與歸納的原則. 進一步探討不變集合中碎形的特點、碎形的產生方法、碎形的度量、碎形壓縮及碎形的藝術. 進一步預測系統運作過程中的無序或無規性的度量. 最後,學習對於任一集合經由疊代後如何產生之不確定性,輔以機率空間之角度來計算次可加潛能量利用其技巧應用到拓樸壓, 更期望證出變分原理(Variational principle). 資料壓縮是資訊科學領域中的一門重要學問, 局部熵為我們提供了一個估算資料潛能量的方式, 而事實上, 我們也可將它想成壓縮的理論下界。也就是說, 我們使用各種壓縮法, 在最理想的情況下能把資料壓縮到等同於它的entropy 的容量. 在實際的應用上, 我們將會發現即便是最好的壓縮方式也只能盡量逼近entropy 大小的境界, 所以entropy 是一個理論上能壓縮到的最小值。 At first, the goal of this project is to study the paper “Invariance entropy for topological semigroup actions” by Colonius, Fukuoka and Santana. This quantity is an invariant factor under topological conjugacy and power rule can be shown. One example shows that the lower entropy dimension can attain any value in [0,1], and the other indicates that the lower and upper entropy dimensions and that in the sense of Bowen can be different. Further research on the characteristics of fractal shape, the method of fractal generation, the measurement of fractal shape, the art of fractal compression and fractal shape in invariant set, and further forecast the measurement of disorder or randomness in the process of system operation. Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension can't be minimal. Moreover, learning how to generate the uncertainty for any set after iteration, supplemented by the perspective of probability space to calculate the sub-potential energy can be used to apply its potential to the topological pressure, but also expect to prove variational principle. Variational principle Compression is an important science in the field of information science, and local entropy provides us with a way to estimate the latent energy of the data, and in fact we can think of it as a theory of compression. In other words, we use a variety of compression methods, in the best case can be compressed to the equivalent of the capacity of its entropy.In practical applications, we will find even the best compression can only try to entropy size approximation of the realm, so entropy is a theory can be compressed to the minimum.
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