Timeline for Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

5 events

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Jan 4, 2012 at 3:04	history	edited	Pengfei	CC BY-SA 3.0	improved formatting
Jan 4, 2012 at 2:55	comment	added	Pengfei		Then I find the following statement in the beginning of Section II.6 of Mane's book: Let $X$ be a compact metric space and $T: X\to X$ a measurable map. $\cdots$ When $\mathcal{M}_f(X)\neq\emptyset$ it is reasonable to ask whether it contains any ergodic maps (typo, should be measures). The answer is yes, and it follows from the results in this section. -----Theorem 6.4 there is about the ergodic decomposition of invariant measure (if exist). Assuming this theorem, then your quesion (1) is true for the even general map $f$.
Jan 4, 2012 at 2:48	comment	added	Pengfei		Another characterization of ergodic decomposition is <br> $\mu(E)=\int_{E(T)}m(E)d\tau(m)$ for every Borel subset $E\subset \widetilde{X}$. See the remark between Theorem 6.4 and Corollary 6.5 in the book "Ergodic Theory and Differentiable Dynamics" by Mane. Indeed Corollary 6.5 is exactly the case we used here.
Jan 3, 2012 at 13:41	comment	added	André Caldas		How do you prove that $\mu(X) = 1$ implies $m(X) = 1$ for $\tau$-a.e. $m$? Remember that having an "ergodic decomposition" means that for every continuous $\phi: \widetilde{X} \to \mathbb{R}$, $\int \phi d\mu = \int_{E(T)} \int \phi dm d\tau$. Also, notice that it might happen that $X \neq T^{-1}X$.
Jan 2, 2012 at 14:49	history	answered	Pengfei	CC BY-SA 3.0