Timeline for Contractibility of connected holomorphic dynamics?
Current License: CC BY-SA 3.0
12 events
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| Nov 19, 2013 at 19:22 | comment | added | Lasse Rempe | If by "holomorphic function" you mean "entire function", then the answer is yes (but this is really a different question). It may not be quite as trivial for transcendental maps as for polynomials, but is still true. | |
| Nov 19, 2013 at 11:16 | comment | added | Sebastien Palcoux | @LasseRempe-Gillen : ok, thank you! Is it true that the non-escaping set of every holomorphic function, has trivial fundamental group (by using your trivial standard results) ? | |
| Nov 18, 2013 at 19:45 | comment | added | Lasse Rempe | Yes, they always have trivial fundamental group. This is trivial using standard results. After giving some thought to the matter, I believe, however, that Cremer quadratic Julia sets are quite unlikely to be path-connected. | |
| Nov 18, 2013 at 18:20 | comment | added | Sebastien Palcoux | My definition of $1$-connected is : with a trivial fundamental group and $0$-connected (i.e. path-connected). Because you don't know whether Cremer Julia sets can be $0$-connected, I guess you don't have the same definition. Anyway, you say that these sets have a trivial fundamental group. | |
| Nov 18, 2013 at 17:37 | comment | added | Lasse Rempe | @Sebastien A Cremer Julia set has no interior and can never disconnect the plane, so - if I have my definitions right - every such Julia set is 1-connected. | |
| Nov 18, 2013 at 11:17 | comment | added | Sebastien Palcoux | @LasseRempe-Gillen : I thought you prove that it's not $1$-connected (because $1$-connected is weaker than contractible). But, in fact, you prove directly that it's not contractible. Thank you ! If $0$-connected Cremer filled Julia sets exist, do you know if they could be not $1$-connected ? | |
| Nov 18, 2013 at 10:52 | vote | accept | Sebastien Palcoux | ||
| Nov 18, 2013 at 6:01 | history | edited | Lasse Rempe | CC BY-SA 3.0 |
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| Nov 18, 2013 at 5:48 | comment | added | Lasse Rempe | @Alex - I will clarify. Sebastien - I don't understand. I believe contractibility is a stronger property than being 1-connected. In any case, I am not sure that it is known whether Cremer Julia sets can be pathwise connected. | |
| Nov 17, 2013 at 21:35 | comment | added | Alexandre Eremenko | In the entire transcendental case, are you sure that the non-escaping set is connected? (I did not understand your "hence", because the non-escaping set contains much more than the Fatou component). | |
| Nov 17, 2013 at 21:10 | comment | added | Sebastien Palcoux | Thank you for this outline ! I'm not sure to well understand about the quadratic Cremer polynomials: do they disprove the weaker property "$0$-connected $\Rightarrow$ $1$-connected"? (If no, is this property still open in the entire case?) | |
| Nov 17, 2013 at 18:17 | history | answered | Lasse Rempe | CC BY-SA 3.0 |