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The parameter space of cubic laminations with a fixed critical leaf
Ergodic Theory and Dynamical Systems. 2016. Vol. 37. P. 2453–2486.
Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.
Blokh A., Oversteegen L., Selinger N. et al., Arnold Mathematical Journal 2026 Vol. 12 No. 1 P. 60–110
We describe a model for the boundary of the connectedness locus of the parameter space of cubic symmetric polynomials. We show that there exists a monotone continuous function from the connectedness locus to the model which is a homeomorphism if the former is locally connected. ...
Added: May 13, 2026
Blokh A., Oversteegen L., Timorin V. et al., Nonlinearity 2025 Vol. 38 No. 7 Article 075014
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 ...
Added: June 12, 2025
Blokh A., Oversteegen L., Selinger N. et al., Ergodic Theory and Dynamical Systems 2025 Vol. 45 No. 8 P. 2314–2340
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 ...
Added: January 7, 2025
Kochetkov Y., / Series arXiv.org e-print archive "arXiv.math". 2024. No. 2401.11208.
A cubic Galois polynomial is a cubic polynomial with rational coefficients that defines a cubic Galois field. Its
discriminant is a full square and its roots $x_1,x_2,x_3$ (enumerated in some order) are real. There exists (and only one) quadratic polynomial $q$ with rational coefficients such that $q(x_1)=x_2, q(x_2)=x_3, q(x_3)=x_1$. The polynomial $r=q(q) \text{ mod } p$ cyclically permutes roots of $p$ ...
Added: February 5, 2024
Blokh A., Oversteegen L., Shepelevtseva A. et al., Moscow Mathematical Journal 2022 Vol. 22 No. 2 P. 265–294
The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such slices as parameter spaces were studied by S. Zakeri, so we call them Zakeri slices. We ...
Added: May 27, 2022
Blokh A., Cheritat A., Oversteegen L. et al., Nonlinearity 2021 Vol. 34 No. 4 P. 2430–2453
A cubic polynomial with a marked fixed point 0 is called an IS-capture polynomial if it has a Siegel disk D around 0 and if D contains an eventual image of a critical point. We show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component of the polynomial parameter space determined by the rational lamination of ...
Added: April 26, 2021
Blokh A., Oversteegen L., Timorin V., , in: Contemporary Mathematics 744 Dynamics: Topology and Numbers (2020).: United States of America: American Mathematical Society, 2020. Ch. 13 P. 205–229.
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 ...
Added: November 7, 2020
Blokh A., Oversteegen L., Timorin V., Science China Mathematics 2018 Vol. 61 No. 12 P. 2121–2138
Added: November 24, 2018
Blokh A., Oversteegen L., Ptacek R. et al., Proceedings of the American Mathematical Society 2018 Vol. 146 No. 11 P. 4649–4660
Added: August 27, 2018
Blokh A., Oversteegen L., Timorin V., Discrete and Continuous Dynamical Systems 2017 Vol. 37 No. 11 P. 5781–5795
Every plane continuum admits a finest locally connected model. The latter is a locally connected continuum onto which the original continuum projects in a monotone fashion. It may so happen that the finest locally connected model is a singleton. For example, this happens if the original continuum is indecomposable. In this paper, we provide sufficient ...
Added: August 16, 2017
Ptacek R., Blokh A., Oversteegen L. et al., Comptes Rendus Mathematique 2017 Vol. 355 No. 5 P. 590–595
W. Thurston constructed a combinatorial model of the Mandelbrot set M2M2such that there is a continuous and monotone projection of M2M2to this model. We propose the following related model for the space MD3MD3of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3(P,c1,c2)∈MD3, then every point z in the Julia set ...
Added: May 30, 2017
Blokh A., Oversteegen L., Ptacek R. M. et al., Discrete and Continuous Dynamical Systems 2016 Vol. 36 No. 9 P. 4665–4702
Polynomials from the closure of the principal hyperbolic domain of the cubic connectedness locus have some specific properties, which were studied in a recent paper by the authors. The family of (affine conjugacy classes of) all polynomials with these properties is called the Main Cubioid. In this paper, we describe a combinatorial counterpart of the ...
Added: July 6, 2016
Blokh A., Oversteegen L., Ptacek R. et al., Communications in Mathematical Physics 2016 Vol. 341 No. 3 P. 733–749
A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a nonrepelling fixed point, for which no perturbation results into a polynomial with Jordan ...
Added: January 11, 2016
Blokh A., Oversteegen L., Ptacek R. M. et al., Nonlinearity 2014 Vol. 27 No. 8 P. 1879–1897
The connectedness locus in the parameter space of quadratic polynomials is called the Mandelbrot set. A good combinatorial model of this set is due to Thurston. By definition, the principal hyperbolic domain of the Mandelbrot set consists of parameter values, for which the corresponding quadratic polynomials have an attracting fixed point. The closure of the ...
Added: August 25, 2014
Galkin S., Shinder E., / Series math "arxiv.org". 2014. No. 1405.5154.
We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...
Added: May 21, 2014
Timorin V., Blokh A., Oversteegen L. et al., / Series math "arxiv.org". 2013. No. 1305.5788.
According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can ...
Added: October 6, 2013
Timorin V., Oversteegen L., Blokh A. et al., / Series math "arxiv.org". 2013. No. 1305.5798.
We discuss different analogs of the main cardioid in the parameter space of cubic polynomials, and establish relationships between them. ...
Added: October 6, 2013
Timorin V., Blokh A., Oversteegen L. et al., / Series math "arxiv.org". 2013. No. 1305.5799.
A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan ...
Added: October 6, 2013