具有控制分解的C~1微分同胚沿不稳定叶层的熵公式
发布时间：2018-10-22 11:12
【摘要】：本论文主要研究了具有控制分解的C1微分同胚沿不稳定叶层的熵与Lyapunov指数的关系,揭示了在"C1 +控制分解"条件下不同层次的Lyapunov指数对相应层次的熵的贡献.论文主要包含两部分内容.在第一部分,针对定义在紧致无边的Riemann流形M上具有控制分解的C1微分同胚f,给出了其对一个不变测度μ而言的沿着第i层不稳定叶层的熵hμi(f)的上界估计.具体来说,得到了如下不等式其中λ1(x)λ2(x)…λu(x)(x)是x点处的正Lyapunov指数,mj(x)是λj(x)的重数,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,针对满足某种绝对连续条件的不变测度μ,进一步得到了hμi(f)的下界估计,从而得到了熵公式.具体来说,若对μ-a.e.x∈Γi以及任意一个从属于Wi的可测分割ζi,我们有(?),其中{μxζi}是与ζi相对应的条件测度族,而λxi是Wi(x)上相应的Riemann测度,则我们给出了下面的公式.
[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，
本文编号：2286994
[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，
本文编号：2286994
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