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:

- weakly.
- , .