Models for spaces of dendritic polynomials

Blokh, Alexander; Oversteegen, Lex G.; Ptacek, Ross; Timorin, Vladlen

doi:10.1090/tran/7482

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…Recall that a dendrite is a locally connected continuum that contains no Jordan curves. A q-lamination with no infinite gaps gives rise to a topological Julia set which is a dendrite; we call such q-laminations dendritic (see [BOPT17,BOPT19]). We will also need Theorem 2.19 from [BOSTV1].…”

Section: Theorem 44 ([Kiw02]mentioning

confidence: 99%

Lavaurs algorithm for cubic symmetric polynomials

Blokh¹,

Oversteegen²,

Selinger³

et al. 2022

Preprint

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To investigate the degree d connectedness locus, Thurston studied σ d -invariant laminations, where σ d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f c (z) = z 2 +c. In the same spirit, we consider the space of all cubic symmetric polynomials f λ (z) = z 3 +λ 2 z in three articles. In the first one we construct the lamination C s CL together with the induced factor space S/C s CL of the unit circle S. As will be verified in the third paper, S/C s CL is a monotone model of the cubic symmetric connectedness locus, i.e., 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 constructing C s CL analogous to the Lavaurs algorithm for constructing a combinatorial model M comb 2 of the Mandelbrot set M 2 .

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Section: Theorem 44 ([Kiw02]mentioning

confidence: 99%

Lavaurs algorithm for cubic symmetric polynomials

Blokh¹,

Oversteegen²,

Selinger³

et al. 2022

Preprint

Self Cite

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show abstract

“…The connectedness locus M d , i.e., the set of all such polynomials with connected Julia sets, has been extensively studied for the last 40 years. Major progress has been made for d = 2 but much less is known for d > 2 (see [BOPT17,BOPT19,Thu19]). Thurston [Thu85] introduced geometric invariant laminations as a way to provide models for connected Julia sets and a model for M 2 .…”

Section: Introductionmentioning

confidence: 99%

Untitled

2023

Conform. Geom. Dyn.

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To investigate the degree d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3-137] studied σ d -invariant laminations, where σ d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f (z) = z 2 +c. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials f λ (z) = z 3 + λ 2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s CL together with the induced factor space S/C s CL of the unit circle S. As will be verified in the third paper of the series, S/C s CL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

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Symmetric cubic laminations

Blokh¹,

Oversteegen²,

Selinger³

et al. 2023

Conform. Geom. Dyn.

Self Cite

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To investigate the degree d d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

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Models for spaces of dendritic polynomials

Cited by 3 publications

References 19 publications

Lavaurs algorithm for cubic symmetric polynomials

Lavaurs algorithm for cubic symmetric polynomials

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Symmetric cubic laminations

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