【摘要】：本論文主要研究了具有控制分解的C1微分同胚沿不穩(wěn)定葉層的熵與Lyapunov指數(shù)的關(guān)系,揭示了在"C1 +控制分解"條件下不同層次的Lyapunov指數(shù)對相應(yīng)層次的熵的貢獻(xiàn).論文主要包含兩部分內(nèi)容.在第一部分,針對定義在緊致無邊的Riemann流形M上具有控制分解的C1微分同胚f,給出了其對一個(gè)不變測度μ而言的沿著第i層不穩(wěn)定葉層的熵hμi(f)的上界估計(jì).具體來說,得到了如下不等式其中λ1(x)λ2(x)…λu(x)(x)是x點(diǎn)處的正Lyapunov指數(shù),mj(x)是λj(x)的重?cái)?shù),u(i,x)=u(x)-i+1,Γi是使得u(i,x)0的x的集合.在第二部分,針對滿足某種絕對連續(xù)條件的不變測度μ,進(jìn)一步得到了hμi(f)的下界估計(jì),從而得到了熵公式.具體來說,若對μ-a.e.x∈Γi以及任意一個(gè)從屬于Wi的可測分割ζi,我們有(?),其中{μxζi}是與ζi相對應(yīng)的條件測度族,而λxi是Wi(x)上相應(yīng)的Riemann測度,則我們給出了下面的公式.
[Abstract]:In this paper, the relationship between entropy and Lyapunov exponent of C1-differential homeomorphism along unstable leaf layer with controlled decomposition is studied, and the contribution of Lyapunov exponents of different levels to the entropy of corresponding layers is revealed under the condition of "C1 dominating decomposition". The thesis mainly includes two parts. In the first part, for the C 1 differential homeomorphism defined on a compact boundless Riemann manifold M with controlled decomposition, the upper bound estimates of entropy h 渭 i (f) along the unstable leaf layer of layer I for an invariant measure 渭 are given. Specifically, we obtain the following inequality where 位 1 (x) 位 2 (x). 位 u (x) (x) is the positive Lyapunov exponent, mj (x) at x point is the multiplicity of 位 j (x), u (I x) = u (x) I 1, and 螕 I is the set of x such that u (IP x) 0. In the second part, for the invariant measure 渭 which satisfies some absolute continuity condition, the lower bound estimate of h 渭 i (f) is further obtained, and the entropy formula is obtained. Specifically, if we have (?) for 渭-a.e.x 鈭，

本文編號：2286994