從物理測度的遍歷性到拓樸化遍歷性之演進

Abstract

遍歷理論是研究一個系統長期行為的數學分析，其中一項廣為人知的結果是：在經過長時間後，時間平均將會趨近空間平均。此事實在動力系統扮演極重要的角色。在本篇論文中，我們將把大部分的探討重點放在緊緻測度空間的離散時間變換，同時敘述並証明一些相關的定理。其次，我們會拋開物理上的測度，在更一般的拓樸空間討論遍歷理論，隨後給出一些基本且重要的例子，如無理數的平移、字串的流動…等等。研究遍歷理論的大師Karl Petersen教授，在1970年發表的一篇論文中，以巧妙的方法確實建構出一個拓樸強混合但非測度強混合之系統。因此在此文章最後，會再對該例研討，把原文中未給出的細節與證明補上，修正一處小瑕疵並對某些部分提出另法證明，這也是本篇最主要的菁華。

Ergodic theory is the mathematical study of the long-term average behavior of dynamical systems. One well-known result is that after a long period of time, the time mean “approaches” the space mean of the system, which is the belief of Thermo Dynamic and the Kinetic Theory of Gases, as assented by Boltzmann. The fact plays a very important role in dynamical systems today. In this thesis, we concentrate on a discrete time transformation on a compact space and review some classical theory, and then we shall study abstract notions of ergodic theory on topological spaces with/without measure and give some basic but important examples, such as irrational translation with measure and symbolic flow with/without measure (Borel). In a paper [1] of Professor Karl Petersen in 1970, he constructed a topologically strong mixing but not measure strong mixing system with artful method. In the last chapter of this presentation, we will study this Petersen system with detail proofs to clarify some of the statements and justifications of some missed particulars.