具覆蓋關係動態函數之高維度擾動的拓樸混沌

Abstract

本論文主要研究高維度系統的拓樸動態，其中系統是擾動由F(x,y)=(f(x),g(x,y))形式之系統且滿足低維度函數f為一連續函數。首先我們會證明如果當低維度函數f具有返回擴張固定點，其微小的C^{1}擾動同樣具有返回擴張固定點。 假設函數g具有局部抑制的區域且系統沿著一連續的參數群{F_{λ}}滿足F_{0}=F。我們會證明如果當低維度函數f為一維度函數且具有正的拓樸熵或f為一高維度函數具有返回擴張固定點，則對於所有夠小的參數λ，F_{λ}也會具有正的拓樸熵。並且我們證明如果當f為一微分同胚具有topologically crossing homoclinic point時，則對於參數λ夠接近0時，F_{λ}具有正的拓樸熵。 更進一步地，我們證明當f具有由轉移矩陣A決定的覆蓋關係時，則F的任意微小C^{0}擾動系統會存在一緊緻正向的不變集且當系統限定在此不變集上時會拓樸半共軛到由A生成的單邊有限型子轉移。此外，如果覆蓋關係滿足strong Liapunov condition且函數g為一壓縮函數，則我們會證明出F的任意微小C^{1}擾動同胚會存在一緊緻的不變集且當系統限定在此不變集上時會拓樸共軛到由A生成的雙邊有限型子轉移。

In this dissertation, we investigate topological dynamics of high-dimensional systems which are perturbed from a continuous map f of the following form F(x,y) = (f(x),g(x,y)). First, we show that if the lower dimensional map f has a snap-back repeller, then the small C^{1} perturbation of f also has a snap-back repeller. Assume that g is locally trapping and the system is along a one-parameter continuous family {F_{λ}} such that F_{0} = F. We show that if f is a one dimensional map and has positive entropy, or f is a high-dimensional map and has a snap-back repeller then F_{λ} has a positive topological entropy for all small parameter λ. Also, we show that if f is a C^{1} diffeomorphism having a topologically crossing homoclinic point, then F_{λ} has positive topological entropy for all λ close enough to 0. Moreover, we show that if f has covering relations determined by a transition matrix A, then any small C^{0} perturbed system of F has a compact positively invariant set restricted to which the perturbated system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C^{1} perturbed homeomorphism of F has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A.