Metric entropy

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…

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: $latex h_{\mu}(T)=\frac{1}{n}h_{\mu}(T^n)$. 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 $la…