多維移位系統的混合性及遍歷理論

Abstract

混合性(mixing property)在動態系統及統計物裡中扮演重要的角色。在 多維度時，很多重要的特性和結果必須藉由各種不同的混合性質；如拓樸 混合topological mixing 和強不可約性strong irreducibility來獲得。然而，檢 驗多維系統是否滿足混合性質是困難的，甚至是shifts of finite type，關於檢 驗方法的相關文獻也相當稀少。對於二維shifts of finite type，我們先前引進 角落擴張(corner-extendable condition)和補洞(hole-filling)的概念，以及不變 對角週期(invariant diagonal cycle)和交換素週期(commutative primitive cycle) 方法。接著再利用轉移矩陣(transition matrix)和連繫矩陣去表示上述概念與 方法。進而得到各種不同混合性質可檢查的充分條件。在此基礎上，我們 希望去進一步研究混合性質。 當混合性的具體條件建立之後，接著下來就要去考慮是否如一維有 natural measure存在。目前較可能的線索是在強不約性之下，會有natural measure存在。若能如此，則Ruelle的整套Thermodynamic formalism就有可 能在此類系統建立。

Mixing property plays important roles in dynamical system and statistical physics. In multi-dimensional system, it is hard to verify the various mixing properties, from topological mixing to strong specifications. Previously, in shift of finite type (SFT), we introduce corner-extendable conditions and hole-filling condition to establish some mixing theorems. The concepts can be expressed in terms of transition matrices and connecting operators. Based on these works, we are going to study the existence of natural measure in systems with strong mixing properties. In the cases that natural measure exists, the theory of Ruelle’s thermodynamic formalism applies.