Equidistribution of critical points of the multipliers in the quadratic family

Firsova, Tanya; Gorbovickis, Igors

doi:10.1016/j.aim.2021.107591

Cited by 1 publication

(7 citation statements)

References 12 publications

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“…Figure 1 provides a numerical approximation of the accumulation set X. We also note that the last part of Theorem A complements the following result, previously obtained by the authors in [5]. THEOREM 1.2.…”

Section: Introduction Consider the Family Of Quadratic Polynomialssupporting

confidence: 79%

“…THEOREM 1.1. [5] The sequence of probability measures 1 #X n x∈X n δ x converges to the equilibrium measure μ bif in the weak sense of measures on C as n → ∞.…”

Section: Introduction Consider the Family Of Quadratic Polynomialsmentioning

confidence: 99%

“…THEOREM 1.2. [5] If c ∈ C \ M is a critical point of some multiplier, then c ∈ X. Equivalently, the following identity holds:…”

Section: Introduction Consider the Family Of Quadratic Polynomialsmentioning

confidence: 99%

“…In particular, this motivates the study of the asymptotic behavior of the critical points of period n multipliers as . In [5], the current authors approached this question from the statistical point of view and proved that the critical points of the period n multipliers equidistribute on the boundary of the Mandelbrot set as .…”

Section: Introductionmentioning

confidence: 99%

“…We recall that a periodic orbit is called primitive parabolic if its multiplier is equal to . As discussed in [5], for every and every periodic orbit of that is not primitive parabolic, the multiplier of this periodic orbit can be viewed as a locally analytic function of the parameter c in the neighborhood of . We denote this function by .…”

Section: Introductionmentioning

confidence: 99%

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Accumulation set of critical points of the multipliers in the quadratic family

Firsova¹,

Gorbovickis

2022

Ergod. Th. Dynam. Sys.

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A parameter $c_{0}\in {\mathbb {C}}$ in the family of quadratic polynomials $f_{c}(z)=z^{2}+c$ is a critical point of a period n multiplier if the map $f_{c_{0}}$ has a periodic orbit of period n, whose multiplier, viewed as a locally analytic function of c, has a vanishing derivative at $c=c_{0}$ . We study the accumulation set ${\mathcal X}$ of the critical points of the multipliers as $n\to \infty $ . This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper, we prove that the accumulation set ${\mathcal X}$ is bounded, connected, and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set ${\mathcal X}$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends c to the Hausdorff dimension of $f_{c}$ does not have critical points outside of the accumulation set ${\mathcal X}$ .

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Section: Introduction Consider the Family Of Quadratic Polynomialssupporting

confidence: 79%

“…THEOREM 1.1. [5] The sequence of probability measures 1 #X n x∈X n δ x converges to the equilibrium measure μ bif in the weak sense of measures on C as n → ∞.…”

Section: Introduction Consider the Family Of Quadratic Polynomialsmentioning

confidence: 99%

“…THEOREM 1.2. [5] If c ∈ C \ M is a critical point of some multiplier, then c ∈ X. Equivalently, the following identity holds:…”

Section: Introduction Consider the Family Of Quadratic Polynomialsmentioning

confidence: 99%