Metric entropy（一）：Ruelle 不等式、Pesin 公式与 Lyapunov 指数

旧博客原文

原题：Metric entropy 1

Some basic thing, include the definition of metric entropy is introduced in my early blog.

Among the other thing, there is something we need to focus on:

1.Definition of metric entropy, and more general, topological entropy.

2.Spanning set and separating set describe of entropy.

3.amernov theorem:

.

Now we state the result of Margulis and Ruelle:

Let be a compact riemannian manifold, is a diffeomorphism and is a -invariant measure.

Entropy is always bounded above by the sum of positive exponents;i.e.,

Where is the multiplicity of and .

Pesin show the inequality is in fact an equality if and is equivalent to the Riemannian measure on . So this is also sometime known as Pesin’s formula.

F.Ledrappier and L.S.Young generate the result of Pesin.

One of their main result is:

is a diffemoephism, where is a compact riemanian manifold, f is compatible with the Lesbegue measure on , and

If and only if on the canonical defined quation manifold $M/W_{\mu}$, i.e. the manifold mod unstable manifold $W_{\mu}$, the induced conditional measure is absolute continuous.

Remark: according to my understanding, the equality just mean in some sense we have the inverse estimate:

This result maybe just mean near the fix point of ,i.e. the place charge the topology of the foliation, we have the inverse estimate. Such a inverse estimate will lead a control of the singularity of the push forward measure on the quation manifold.  So have good regularity. But this idea is not complete to solve the problem.

Now we begin to get a geometric explain and which will lead a rigorous proof of the inequality:

At first we could observe that the long time average of could be diagonal. Assume after diagonal the eigenvalue is

.

This eigenvalue could divide into 3 parts: <0,=0,>0.

This will lead to a direct sum decomposition of the tangent bundle :

Where $E_u$ is the part corresponding to the eigenvalue>0, For this part we consider the more refinement decomposition:

, is the eigenvector space of . The dimension of $V_k$ is $dim V_k$.

On the other hand, we have a equality of metric entropy:

.

For the later one, is a measurable partition of , then could always be refine to a smaller partition , and we have:

.

Now we arrive the central place of the proof:

every partition could be refine by a partition with boundary of almost all cubes is parallel to the foliation. So  we focus ourselves on the portion and all boundary of cubes in is parallel to the eigenvector.

Under this situation, we need only estimate the numbers of . Estimate it is not very difficult. we need only observe the following two thing:

1.

exists a.e. in . So this lead to the definition of foliation almost everywhere, and except a measurable zero set. In fact this set is the set of fix point of under .

    2.

After a rescaling, every point which is not a fix point of could be understand as it is far away from fix points. Then the foliation could be understand as  a product space locally. The flow with the direction which the eigenvalue is less than 1 cold not change . The direction with eigenvalue equal to 1 is just transition and just change the number of with polynomial growth. But the central thing is the direction with eigenvalue large than one and will make change with viscosity . and we product it and get :

.

In fact the proof only need to be

 

 

---

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

metric entropy 衡量一个保测动力系统在可测意义下产生信息的速率。对光滑动力系统，它和 Lyapunov exponents 之间有深刻联系：正 Lyapunov 指数给出不稳定方向上的体积增长，而 entropy 记录可区分轨道的增长。

1. Entropy 的基本图像

给定有限 measurable partition $\mathcal P$，Shannon entropy 是

$$H_\mu(\mathcal P)=-\sum_{A\in\mathcal P}\mu(A)\log\mu(A).$$

迭代下的平均信息增长定义为

$$h_\mu(f,\mathcal P)=\lim_{n\to\infty}\frac1n H_\mu\left(\bigvee_{j=0}^{n-1}f^{-j}\mathcal P\right).$$

再对所有 partition 取上确界，得到 $h_\mu(f)$。

2. Spanning 与 separating

拓扑 entropy 可以用 $(n,\varepsilon)$-spanning set 或 separating set 描述。两个点若在前 $n$ 次迭代中始终很近，就被看作同一条轨道影子。entropy 记录为了覆盖所有轨道影子，需要多少个名字。

这个图像和 metric entropy 的 partition 定义相互对应：一个是拓扑尺度，一个是测度尺度。

3. Ruelle inequality

设 $f$ 是紧 Riemannian manifold 上的 $C^1$ diffeomorphism，$\mu$ 是 $f$-invariant measure。Ruelle 不等式说

$$h_\mu(f)\le \int \sum_{\lambda_i(x)>0}\lambda_i(x)m_i(x)\,d\mu(x).$$

右端是正 Lyapunov exponents 的总和。直观上，系统能产生的信息不可能超过不稳定方向的体积膨胀能力。

4. Pesin formula

在更光滑并且测度与 Riemannian volume 绝对连续的情形，Pesin 公式给出等号：

$$h_\mu(f)=\int \sum_{\lambda_i(x)>0}\lambda_i(x)m_i(x)\,d\mu(x).$$

这说明所有不稳定方向上的 expansion 都真正转化成了信息增长，没有被 singular conditional measures 损失掉。

5. Ledrappier-Young 的视角

Ledrappier-Young 理论进一步解释等号何时成立：关键在于 unstable foliation 上的 conditional measures 是否绝对连续。若条件测度太奇异，几何膨胀不一定变成可测 entropy；若条件测度足够连续，膨胀就能被 entropy 读出来。

因此 entropy、Lyapunov exponents 和 foliation 上的条件测度，实际上是同一件事的三个侧面。