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Abstract As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so calledquadratic minor laminationof Thurston. In the same spirit, we consider the space of allcubic symmetric polynomials$$f_\unicode{x3bb} (z)=z^3+\unicode{x3bb} ^2 z$$in three articles. In the first one, we construct thecubic symmetric comajor laminationtogether with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of thecubic symmetric connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination.
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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
- 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
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/conm/744
