Timeline for What do convergent sequences of rational functions look like?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 19, 2021 at 13:06 | comment | added | Alexandre Eremenko | @Asvin: thanks for the reference. When I wrote "must be known" I meant this paper, just forgot the author and the title. | |
| Jul 18, 2021 at 20:28 | comment | added | Asvin | This paper (link.springer.com/article/10.1007/BF02392088) by Segal has a result along the lines of what Alexandre seems to be suggesting | |
| Jul 18, 2021 at 20:27 | comment | added | Asvin | @PierrePC That's an interesting question about homotopy types. Do let me know if you find anything! | |
| Jul 18, 2021 at 20:07 | comment | added | Pierre PC | In degree 1, the space is $\mathrm{PGL}_2(\mathbb C)$, which I believe should be diffeomorphic to $\mathrm{SO}_3(\mathbb R)\times\mathbb R$, which indeed isn't compact. I am not sure I follow your argument for the homotopy type, though. Are you saying your argument gives a retract from the space of continuous maps to that of holomorphic ones? | |
| Jul 18, 2021 at 19:44 | comment | added | Alexandre Eremenko | Since for every open discrete map $f$ between two spheres, there exists a homeomorphism $\phi$ such that $f\circ\phi$ is a rational function, the question about homotopy type is purely topological, not analytic. Also notice that the space is not compact, even for degree $1$. Also, notice that the space is not compact, even for degree $1$: the sequence f_n(z)=z/n$ has no limit points. | |
| Jul 18, 2021 at 19:31 | comment | added | Alexandre Eremenko | yes, I am talking about the whole Riemann sphere. Yes, they are connected, though I have no ready reference. Homotopy type I do not know, but I suppose this must be known. | |
| Jul 18, 2021 at 17:38 | comment | added | Pierre PC | @AlexandreEremenko Nice! I assume you are still talking about the Riemann sphere. I think these are connected, so we know the connected components of this space. Drifting away from the OP's question, I would be interested in more results about its homotopy type. I convinced myself, perhaps wrongly, that the degree 1 (and -1) mappings have the homotopy type of $\mathbb{RP}^3$. Of course for the degree 0 we have a copy of $\mathbb{CP}^1$. | |
| Jul 18, 2021 at 13:10 | comment | added | Alexandre Eremenko | Since the degree of a rational function is a continuous function on this space, it splits into infinitely many components, one for each degree. | |
| Jul 18, 2021 at 8:27 | history | edited | Pierre PC | CC BY-SA 4.0 |
Generalised to general complex (metric) manifolds.
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| Jul 18, 2021 at 8:17 | history | answered | Pierre PC | CC BY-SA 4.0 |