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 $latex (X,T,\mu)$, where $latex \mu$ is a $latex T-$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:

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

We define the entropy at infinity:

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

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

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 $latex \mathbb N^{\mathbb N},\sigma$, 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 $latex C^{\infty}$ 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

$latex (X,d)$ metric space

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

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

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


Kapok theorem:

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

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

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

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



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

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

Simplified entropy formula:

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

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

Simplified entropy inequality:

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

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

Weak entropy dense:

$latex M_e(X,T)$ is weak entropy dense in $latex M(X,T)$. $latex \forall \lambda>0$, $latex \forall \mu\in M(X,T)$, $latex \exists \mu_n\in M_e(X,T)$, satisfied:

  1. $latex \mu_n\to \mu$ weakly.
  2. $latex h_{\mu_n}(T)>h_{\mu}(T)-\lambda$, $latex \forall \lambda>0$.