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 , where is a 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:

We define the entropy at infinity:

Where varies in all sequences of measure coverage to in the sense for all , measurable then .

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 , 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 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

metric space

is a continuous map.

, then is still a metric.

Easy to see . This identity could be proved by the cretition of entropy by -seperate set and -cover set.

 

Kapok theorem:

compact, for every ergodic measure the following formula hold:

.

Where is the measure theoretic entropy of .

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

 

 

Let defined the moduli space of -invariant portability measure.

Let defined the moduli space of ergodic -invariant probability measure.

Simplified entropy formula:

satisfied simplified entropy formula if surfaced small and , .

.

Simplified entropy inequality:

If suffciently small, , .

.

Weak entropy dense:

is weak entropy dense in . , , , satisfied:

1. weakly.
2. , .

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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。

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。