Timeline for Is there any elementary proof of No wandering domain for polynomials
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 21, 2015 at 9:43 | vote | accept | yaoxiao | ||
| Jan 21, 2015 at 9:44 | |||||
| Dec 25, 2014 at 13:44 | comment | added | Adam Epstein | This treatment avoids the Ahlfors-Bers discussion entirely. However, it does rely on the Bers-Lakic Approximation Theorem: for any closed $E\subseteq\widehat{\mathbb{C}}$, the linear space of meromorphic quadratic differentials on $\widehat{\mathbb{C}}$ whose poles are all simple and elements of $E$ is $L^1$-dense in the space of integrable quadratic differentials on $\widehat{\mathbb{C}}$ which are holomorphic outside $E$. My feeling is that McMullen's infinitesimal argument (as presented by Zakeri) is softer yet. | |
| Dec 25, 2014 at 13:37 | comment | added | Adam Epstein | I'm afraid it isn't. It is not difficult to reconstruct the proof by starting with Sullivan's, but using Beltrami differentials to parametrize infinitesimal deformations rather than actual (quasiconformal) deformations. For example, you could take Sullivan's proof and consider how the individual steps interact with the operation of taking the tangent vector field to a smooth path of deformations. Once this has been done, the discussion becomes linear (functional analysis) and the the dual result in the space of quadratic differentials may be obtained by first principles. | |
| Dec 24, 2014 at 16:16 | comment | added | yaoxiao | Thank you, professor Adam Epstein. Where can I you qudratic differentials proof? Does it publish or not? | |
| Dec 23, 2014 at 16:24 | comment | added | Adam Epstein | In fact, only infinitesimal deformation is involved in that argument, which is actually McMullen's. I have a different presentation in the dual space of quadratic differentials. While it is fair to say that these are inspired by Teichmuller spaces, none of the usual machinery (e.g Ahlfors-Bers Theorem) is required. | |
| Dec 23, 2014 at 8:04 | comment | added | Lasse Rempe | This is really still Sullivan's proof, however. In other words, it still uses quasiconformal deformations and (crucially) finite-dimensionality of the parameter space. | |
| Dec 21, 2014 at 18:23 | comment | added | Alexandre Eremenko | "Some special cases" are hyperbolic cases. If no hyperbolicity assumptions are made, one has to use Teichmuller spaces, in one way or another. | |
| Dec 21, 2014 at 16:38 | comment | added | yaoxiao | Thank you, I know this elegant proof. In some sense, the back ground of Teichmuller space is still needed. Someones have used technique of hyperbolic metric in some special class. | |
| Dec 21, 2014 at 16:31 | history | answered | Igor Rivin | CC BY-SA 3.0 |