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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 <blank>' 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 |