Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs

Benoit B. Mandelbrot
Additional contact information
Benoit B. Mandelbrot: Yale University, Mathematics Department

Chapter C10 in Fractals and Chaos, 2004, pp 117-124 from Springer

Abstract: Abstract Within the M-set of the map z → λz(1-z), consider a sequence of points λm having a limit point λ. Denote the corresponding F* -sets by ℱ*(λm) and ℱ*(λ). In general, lim ℱ*(λm) = ℱ*(lim λm), but there is a very important exception. In some cases, the sets ℱ*(λm) do not converge to either a curve or a dust, but converge to a domain of the A -plane, part of which is called the Siegel disc l while the rest is made of the preimages of ℒ. In such cases, ℱ*(lim λm) is not the set lim ℱ*λm but only that set’s boundary. The intuitive meaning of this behavior is discussed and illustrated in terms of the so-called Peano curves, and a mathematical question is raised concerning the nonrational and non-Siegel λ.

Keywords: Double Point; Jordan Curve; Golden Ratio; Continue Fraction Expansion; Fibonacci Sequence (search for similar items in EconPapers)
Date: 2004
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4757-4017-2_10

Ordering information: This item can be ordered from
http://www.springer.com/9781475740172

DOI: 10.1007/978-1-4757-4017-2_10

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().