分类： Metric entropy

Metric entropy（二）：entropy map 的上半连续性与 infinity entropy
旧博客原文

原题：Metric entropy 2

I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”.

Now we focus on the upper semi-continuouty of the metric entropy map. The object we investigate is $(X,T,\mu)$ , where $\mu$ is a $T-$ invariant measure.

The insight to make us interested to this kind of problem is a part of variational problem, something about the existence of certain object which combine a certain moduli space to make some quantity attain critical value(maximum or minimum). The most simple example maybe Isoperimetric inequality and Dirichlet principle of Laplace. Any way, to establish such a existence result a classical approach is to proof the upper semi-continuouty and bounded for associate energy of the problem. In our case the semi-continuouty will be some thin about the regularity of the entropy map:

$E:M(X,T)\to h_{\mu}.$

We define the entropy at infinity:

$sup_{(\mu_n)}limsup_{\mu_n\to 0}h_{\mu_n}(T)$

Where $(u_n)_{n=1}^{\infty}$ varies in all sequences of measure coverage to $0$ in the sense for all $A\subset M$ , $A$ measurable then $\lim_{n\to \infty} \mu_{n}(A)=0$ .

Compact case

we say some thing about the compact case, In this case we have finite partition with smaller and smaller cubes, this could be understand as a sequences of smaller and smaller scales. A example to explain the differences is $\mathbb N^{\mathbb N},\sigma$ , shift map on countable alphabet.

Because of this thing, there is a good sympolotic model, i.e. h-expension, and it generalization asymptotically h-expension equipped on a compact metric space $X$ have been proved to be that the corresponding entropy map is upper semi-continous.

In particular $C^{\infty}$ diffeomorphisms on compact manifold is asymptotically h-expensive.

Natural problem but I do not understand very well:

Why it is natural to assume the measure to be probability measure in the non-compact space?

Non-compact case

$(X,d)$ metric space

$T:X\longrightarrow X$ is a continuous map.

$d_n(x,y)=\sup_{0\leq k\leq n-1}d(T^kx,T^ky)$ , then $d_{n}$ is still a metric.

Easy to see $\frac{1}{n}h_{\mu}(T^n)=h_{\mu}(T)$ . This identity could be proved by the cretition of entropy by $\delta$ -seperate set and $\delta$ -cover set.

Kapok theorem:

$X$ compact, for every ergodic measure $\mu$ the following formula hold:

$h_{\mu}(T)=\lim_{\epsilon \to 0}limsup_{n\to \infty}\frac{1}{n}logN_{\mu}(n,\epsilon,\delta)$ .

Where $h_{\mu}(T)$ is the measure theoretic entropy of $\mu$ .

Riquelme proved the same formula hold for Lipchitz maps on topological manifold.

Let $M_e(X,T)$ defined the moduli space of $T$ -invariant portability measure.

Let $M_(X,T)$ defined the moduli space of ergodic $T$ -invariant probability measure.

Simplified entropy formula:

$(X,d,T)$ satisfied simplified entropy formula if $\forall \epsilon >0$ surfaced small and $\forall \delta\in (0,1)$ , $\mu\in M _e(X,T)$ .

$h_{\mu}(T)=\limsup_{n\to \infty}\frac{1}{n}log(N_{\mu}(n,\epsilon,\delta))$ .

Simplified entropy inequality:

If $\epsilon>0$ suffciently small, $\mu \in M_{e}(X,T)$ , $\delta\in (0,1)$ .

$h_{\mu}(T)\leq \limsup_{n\to \infty}\frac{1}{n}log(N_{\mu}(n,\epsilon,\delta))$ .

Weak entropy dense:

$M_e(X,T)$ is weak entropy dense in $M(X,T)$ . $\forall \lambda>0$ , $\forall \mu\in M(X,T)$ , $\exists \mu_n\in M_e(X,T)$ , satisfied:
1. $\mu_n\to \mu$ weakly.
2. $h_{\mu_n}(T)>h_{\mu}(T)-\lambda$ , $\forall \lambda>0$ .
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。

entropy map 的上半连续性是变分问题中的关键正则性。若想证明某个 invariant measure 使 entropy 或 pressure 达到最大，通常需要紧性和上半连续性。

entropy map 的上半连续性关系到最大熵测度存在性；非紧情形还要控制 entropy at infinity。

1. Entropy map

给定动力系统 $f:X\to X$，考虑 invariant measures 空间 $\mathcal M_f(X)$ 上的函数

$$\mu\mapsto h_\mu(f).$$

若 $\mu_j\to\mu$ 弱收敛，希望有

$$\limsup_{j\to\infty}h_{\mu_j}(f)\le h_\mu(f).$$

这就是上半连续性。

2. 紧空间情形

在紧空间上，可以用越来越细的有限 partition 近似 entropy。若系统具有 expansiveness 或 asymptotic h-expansiveness，entropy map 常有较好的上半连续性。

3. 非紧空间的困难

非紧空间中，测度可能逃向无穷远。即使 $\mu_j$ 弱收敛到某个极限，entropy 也可能在逃逸部分携带额外信息。这个额外损失可用 entropy at infinity 衡量。

4. Entropy at infinity

粗略地说，entropy at infinity 记录所有逃向无穷远的测度序列可能保留的 entropy：

$$h_\infty=\sup_{\mu_j\to0}\limsup h_{\mu_j}(f).$$

若 $h_\infty$ 小于系统的 topological entropy，就有机会证明最大熵测度存在。

5. 几何动力系统中的意义

在 geodesic flow 中，entropy 与轨道增长、曲率和测地线逃逸相关。上半连续性问题本质上是在问：复杂轨道是否可能全部跑到 cusp 或无穷远处。若不能，就能在内部找到达到最大 entropy 的测度。
2017年11月21日
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:

$h_{\mu}(T)=\frac{1}{n}h_{\mu}(T^n)$ .

Now we state the result of Margulis and Ruelle:

Let $M$ be a compact riemannian manifold, $f:M\to M$ is a diffeomorphism and $\mu$ is a $f$ -invariant measure.

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

$h_{m}(f)\leq \int_{i}\lambda_i^{+}(x)dimE_i(x)dm(x).$

Where $dimE_i(x)$ is the multiplicity of $\lambda_i(x)$ and $a^{+}=max(a,0)$ .

Pesin show the inequality is in fact an equality if $f\in C^2$ and $m$ is equivalent to the Riemannian measure on $M$ . 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:

$f:M\to M$ is a $C^2$ diffemoephism, where $M$ is a compact riemanian manifold, f is compatible with the Lesbegue measure on $M$ , and

$h_m({f,\mu})=\int_{M}\lambda_idim(V_i)dm$

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 $m_{\xi}$ is absolute continuous.

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

$h_{m}(f,\mu)\geq \int_{M}\lambda_idim(V_i)dm.$

This result maybe just mean near the fix point of $f$ ,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 $m_{\xi}$ on the quation manifold. So $m_{\xi}$ 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:

$h_{m}(f)\leq \int_{i}\lambda_i^{+}(x)dimE_i(x)dm(x).$

At first we could observe that the long time average $\lim_{n\to \infty}\frac{1}{n}log||Df^n||$ of $Df$ could be diagonal. Assume after diagonal the eigenvalue is

$\lambda_1\leq \lambda_2\leq \lambda_3\leq...\leq \lambda_{n-1}\leq \lambda_n$ .

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

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

$TM\simeq E_{u}\otimes E_s\otimes E_c.$

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

$E_u=\otimes_{k=1}^rV_k$ , $V_k$ is the eigenvector space of $\lambda_k$ . The dimension of $V_k$ is $dim V_k$.

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

$h_{m}(f)=\frac{1}{n}h_{m}(f^n)=\sup_{\alpha\in partition \ set}\frac{1}{n}h_m(f^n,\alpha)$ .

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

$h_{m}(f,\alpha)\leq h_{m}(f,\beta)$ .

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 $\beta$ and all boundary of cubes in $\beta$ is parallel to the eigenvector.

Under this situation, we need only estimate the numbers of $\vee_{i=1}^nT^i\beta$ . Estimate it is not very difficult. we need only observe the following two thing:

1.

$\lim{n\to \infty}$ exists a.e. in $M$ . 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 $M$ under $f$ .

2.

After a rescaling, every point which is not a fix point of $f$ 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 $\vee_{i=1}^nT^i\beta$ . The direction with eigenvalue equal to 1 is just transition and just change the number of $\vee_{i=1}^nT^i\beta$ with polynomial growth. But the central thing is the direction with eigenvalue large than one and will make $\vee_{i=1}^nT^i\beta$ change with viscosity $e^{\lambda_i}$ . and we product it and get :

$h_{m}(f,\mu)\leq \int_{M}\lambda_i dim(V_i)dm$ .

In fact the proof only need $f$ to be $C^1$

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。

Ruelle 不等式和 Pesin 公式把 entropy 与正 Lyapunov 指数联系起来。

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。

2017年11月21日