Metric entropy

I am reading the article “ENTROPY THEORY OF GEODESIC FLOWS”. Now we focus on the upper semi-continuity of the metric entropy map. The object we investigate is $(X,T,\mu)$, where $\mu$ is a $T$-invariant measure. The insight that makes us interested in this kind of problem is a part of a variational problem, something about the existence of a certain object that combines a certain moduli space to make some quantity attain a critical value (maximum or minimum). The simplest example may…

Some basic thing, including the definition of metric entropy, is introduced in my early blog. Among the other thing, there is something we need to focus on: 2. Spanning set and separating set describe of entropy. 3.amernov theorem: $latex h_{\mu}(T)=\frac{1}{n}h_{\mu}(T^n)$. Now we state the result of Margulis and Ruelle: Let $latex M$ be a compact Riemannian manifold, $latex f: M\to M$ is a diffeomorphism and $latex \mu$ is a $latex f$-invariant measure. Entropy is always bounded above by the s…