Both the Mandelbrot set and filled Julia sets are subsets in the complex plane derived by studying iterations of complex polynomials. We develop a matricial framework to establish an alternate form of iteration by complex polynomials using a sequence of affine transformations. Using this framework, we are able to check membership in a filled Julia set and the Mandelbrot set by studying boundedness of sequences of matrices. Specifically, we show that a complex number belongs to the Mandelbrot set if and only if a particular sequence of matrices is bounded in the operator norm, and a complex number belongs to a filled Julia set if and only if a particular sequence of matrices is bounded in operator norm.
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Dynamical generation of parameter laminations
Abstract. Local similarity between the Mandelbrot set and quadratic Julia
sets manifests itself in a variety of ways. We discuss a combinatorial one, in the
language of geodesic laminations. More precisely, we compare quadratic invariant
laminations representing Julia sets with the so-called Quadratic Minor
Lamination (QML) representing a locally connected model of the Mandelbrot
set. Similarly to the construction of an invariant lamination by pullbacks of
certain leaves, we describe how QML can be generated by properly understood
pullbacks of certain minors. In particular, we show that the minors of
all non-renormalizable quadratic laminations can be obtained by taking limits
of “pullbacks” of minors from the main cardioid
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- Award ID(s):
- 1807558
- NSF-PAR ID:
- 10149496
- Date Published:
- Journal Name:
- Contemporary mathematics
- Volume:
- 744
- ISSN:
- 2705-1056
- Page Range / eLocation ID:
- 205-229
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/conm/744
