Timeline for Systems with trivial cohomology
Current License: CC BY-SA 4.0
10 events
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| Feb 8, 2020 at 1:27 | comment | added | YCor | Maybe my confusion comes from the possibility that $\beta C(X)$ might be non-closed. What my argument (if correct) proves is indeed that unique ergodicity means that the closure of $\beta C(X)$ is a hyperplane. | |
| Feb 8, 2020 at 0:56 | comment | added | Veridian Dynamics | I see why my condition implies uniquely ergodicity, but not the converse. Actually, in arxiv.org/abs/1911.07700 the authors prove that some "S-adic systems of rank $K$" are such that $H^1$ has dimension $K$ as vector space, and I know that this kind of systems are uniquely ergodic. So, if I understood everything right, the converse is false. | |
| Feb 7, 2020 at 14:33 | comment | added | YCor | Aren't you precisely asking which systems $(X,T)$ are uniquely ergodic? Uniquely ergodic means that the only positive invariant linear forms on $C(X)$ are positive scalar multiples of "taking the integral" with respect to a given invariant measure. So clearly your condition implies uniquely ergodic, but I was wondering whether the converse also holds. It seems so: if one has a invariant signed measure, its positive and negative mutually singular parts are invariant too. So if this is correct there is plethora of examples beyond irrational rotations. | |
| Feb 7, 2020 at 13:41 | history | edited | Veridian Dynamics | CC BY-SA 4.0 |
added 230 characters in body
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| Feb 7, 2020 at 13:37 | comment | added | YCor | Well, can you edit your question so as to present the setting. I have no idea if you work on the circle or more general spaces/ compact group, and which which restrictions on the homeomorphism. The $H^1$ you define works for an arbitrary self-homeomorphism of a compact space, then can be specified to rotations of the circle. (By the way, $H^1(X)=\mathbf{C}$ is not trivial, it's 1-dimensional. For more general group actions, the space of co-invariants might be 0-dimensional.) | |
| Feb 7, 2020 at 13:36 | comment | added | Veridian Dynamics | My question is: if $(X,T)$ is a topological dynamical system such that $H^1(X)$ is trivial then, is $(X,T)$ conjugated in some sense to $(S^1,+\alpha)$, for some $\alpha$? | |
| Feb 7, 2020 at 13:34 | comment | added | Veridian Dynamics | If $X = (G,\cdot a)$ is a group rotatation such that the orbit $\{a^n:n\in\mathbb{Z}\}$ is dense in $G$, isn't it $G$ minimal? | |
| Feb 7, 2020 at 13:30 | comment | added | YCor | "other minimal group rotations"? but the minimal group rotations are precisely the irrational rotations. What is the question? | |
| Feb 7, 2020 at 13:30 | history | edited | YCor | CC BY-SA 4.0 |
fixed statement
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| Feb 7, 2020 at 13:25 | history | asked | Veridian Dynamics | CC BY-SA 4.0 |