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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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This content will become publicly available on June 12, 2026
A model of the cubic connectedness locus
Abstract We describe a locally connected model of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of the decomposition into points. This model is similar to a quotient of the combinatorial quadratic Mandelbrot set in which all baby Mandelbrot sets, as well as the filled Main Cardioid, are collapsed to points. All fibres of the model, possibly except one, are connected. The authors are not aware of other known models of the cubic connectedness locus.
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- Award ID(s):
- 2349942
- PAR ID:
- 10613017
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 38
- Issue:
- 7
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 075014
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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