Talk:Escaping set
Add topicLatest comment: 5 years ago by Lasserempe in topic Boundary and never closedness
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Boundary and never closedness
[edit]I have removed the following line from the properties section citing[1]:
- The boundary of the escaping set is exactly the Julia set.[1] In particular, the escaping set is never closed.
The first sentence is not true: The Julia set of exp(z) is the whole plane, which is not the boundary of anything. If there is an additional condition that ensures the statement, it should be added. If the second sentence is true for other reasons, it should be readded with a proper citation. I have not found either in the cited source.[1] Hiferator (talk) 15:38, 30 July 2020 (UTC)
- The statement made is correct. The boundary of any subset that is dense but containis no open set is, by definition, the entire space. This is proved in the paper by Eremenko. Indeed, it is formula (1) on the very first page of that paper, which is shown to hold for every entire function (see the second line of the second page, page 340). Lasse (talk) 13:24, 3 August 2020 (UTC)
References
- 1 2 3 Eremenko, A (1987). "On the iteration of entire functions" (PDF). Banach Center Publications, Warsawa, PWN. 23: 339–345.