André Caldas
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 Jul 16 comment System with invariant measure, but no ergodic measure. @JulianNewman: It does not seem to me that Daniel defined his measure over the whole sigma algebra in his first attempt. In his second attempt, there is nothing that ensures the measure $\mu'$ is in fact ergodic. Nov 12 awarded Nice Answer Aug 10 awarded Nice Question Mar 28 awarded Popular Question Nov 18 awarded Nice Question Sep 30 awarded Yearling Apr 4 awarded Necromancer Jan 3 comment Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures. 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 $X \neq T^{-1}X$. Oct 13 revised locally connected versus locally compact Fixed math statement. Oct 13 comment locally connected versus locally compact This is a very neat way to justify my intuitive but not so elaborate and well justified belief that any neighborhood should have the property. Note taken! :-) I think of a neighborhood as a "sufficiently large set". Large enough to be considered a neighborhood. Often, one needs to choose a "small" set, but at the same time, big enough to make the small step big enough to achieve the goal... "path connect two points", for instance. Oct 13 comment Infinite dimensional vector spaces with compact unit ball I liked your question, Leandro! I had never thought of $\mathbb{R}$ as a infinite dimensional space with compact unit ball! You opened my mind a bit! :-) +1 Oct 13 comment locally connected versus locally compact Which one is your "standard 'locally ' definition"? :-) Oct 13 answered locally connected versus locally compact Oct 11 comment Extending open maps to Stone-Cech compactifications The extension is unique, are you just asking if the extension is or not an open surjection? Oct 9 accepted Compact group extension of a zero entropy system. Oct 7 comment Compact group extension of a zero entropy system. @Asaf: Thank you very much for the help. I will study your answer. :-) Oct 5 comment Compact group extension of a zero entropy system. By the way, a similar argument to "Lemma 7" is used in Theorem 2.3 of "On Li-Yorke Pairs": imath.kiev.ua/~skolyada/LY.pdf "Since an isometric extension of a zero-entropy system has still entropy zero [...]" Oct 5 comment Compact group extension of a zero entropy system. @Asaf: Oops... (again!) Sorry for not paying the attention you deserve! Theorem 8.2 reads: If $(X, \mathcal{B}, \mu, T)$ is an ergodic isometric extension of $(Y, \mathcal{D}, \nu, T)$ then $(X, \mathcal{B}, \mu, T)$ is a factor of an ergodic group extension of $(Y, \mathcal{D}, \nu, T)$. In Furstenberg's paper, the definition for isometric extensions is quite complicated, but there is a comment (p. 236) saying: Extensions for which $\mathcal{E}(X|Y,T) = L^2(X)$ will be called isometric extensions. Oct 5 revised Compact group extension of a zero entropy system. Changed the measure to be ergodic. Oct 5 comment Compact group extension of a zero entropy system. @Asaf: I will have to study a bit more to fully understand your comments... :-) I guess I am forgetting the fact that $\mu$ is supposed to be ergodic. I will edit (again!) the post. By the way, I am trying to make sense of the demonstration of Lemma 7 at math.u-psud.fr/~ruette/articles/asympent.pdf