Metric entropy
-
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…
-
Metric entropy 1
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…