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Carsten Lunde Petersen

Publications and source records attributed to Carsten Lunde Petersen.

At least 19 recordsLinked to original sources
arXiv2026

On qc compatibility of satellite copies of the Mandelbrot set: II

The Mandelbrot set $\mathcal{M}$ contains infinitely many small copies of itself, each canonically homeomorphic to $\mathcal{M}$ via the Douady--Hubbard theory of polynomial-like maps. These copies come in two kinds: primitive copies, whose principal hyperbolic component carries a cusp at its root, and satellite copies, whose boundary is smooth at the root. Douady and Hubbard conjectured that the straightening maps of analytic families of polynomial-like maps are quasiregular, predicting that primitive copies are quasiconformally homeomorphic to $\mathcal{M}$ and de-rooted satellite copies to $\mathcal{M}\setminus\{1/4\}$ --- hence that satellite copies are mutually quasiconformally homeomorphic away from their roots. Lyubich proved the primitive case, and showed that satellite copies are quasiconformally homeomorphic to $\mathcal{M}$ outside every neighbourhood of the root. Whether the homeomorphisms between satellite copies are quasiconformal at the roots remained open. In a previous work we gave a negative answer: satellite copies $\mathcal{M}_{p/q}$ and $\mathcal{M}_{p'/q'}$ with $q \neq q'$ are not quasiconformally homeomorphic, disproving the conjecture; and we conjectured that copies whose rotation numbers share the same denominator are quasiconformally homeomorphic. In the present paper we prove that they are. Together, these results yield a complete geometric classification: two satellite copies of $\mathcal{M}$ are quasiconformally equivalent if and only if their rotation numbers have the same denominator. This settles the quasiconformal geometry of the small copies of the Mandelbrot set.

math.DS↗
arXiv2026

Julia Sets for Sequences of Monomials

We study Julia sets arising from non-autonomous iteration of monomials on the Riemann sphere. We give a precise description of the Julia set associated to an arbitrary sequence of monomials in terms of their coefficients and degrees. Combined with the basic invariance relation for non-autonomous Julia sets, this yields a plethora of striking examples in polynomial non-autonomous dynamics: finite Julia sets of arbitrary cardinality, non-trivial Julia sets with non-empty interior, Julia sets that are perfect but not uniformly perfect, and Julia sets with empty interior but positive area. We also describe a connection between autonomous and non-autonomous Julia sets of monomial sequences through Kuratowski limits.

math.DS↗
arXiv2025

Lemon limbs of the cubic connectedness locus

We describe a primary limb structure in the connectedness locus of complex cubic polynomials, where the limbs are indexed by the periodic points of the doubling map $t \mapsto 2t \ (\operatorname{mod} {\mathbb Z})$. The main renormalization locus in each limb is parametrized by the product of a pair of (punctured) Mandelbrot sets. This parametrization is the inverse of the straightening map and can be thought of as a tuning operation that manufactures a unique cubic of a given combinatorics from a pair of quadratic hybrid classes.

math.DS↗
arXiv2025

Roots of polynomial sequences in root-sparse regions

Given a family $(q_k)_k$ of polynomials, we call an open set $U$ root-sparse if the number of zeros of $q_k$ is locally uniformly bounded on $U$. We study the interplay between the individual zeros of the polynomials $q_k$ and those of the $m$th derivatives $q_k^{(m)}$, in a root-sparse open set $U$, as $k\to\infty$. More precisely, if the root distributions $μ_k$ of $q_k$ converge weak* to some compactly supported measure $μ$, whose potential is nowhere locally constant on a root-sparse open set $U$, then we link the roots of the $m$th derivative $q_k^{m}$, for an arbitrary $m>0$, to the roots of $q_k$ and the critical points of the potential $p_μ$ on compact subsets of $U$. We apply this result in a polynomial dynamics setting to obtain convergence results for the roots of the $m$th derivative of iterates of a polynomial outside the filled-in Julia set. We also apply our result in the setting of extremal polynomials.

math.CV↗
arXiv2024

Buff forms and invariant curves of near-parabolic maps

We introduce a general framework to study the local dynamics of near-parabolic maps using the meromorphic $1$-form introduced by X.~Buff. As a sample application of this setup, we prove the following tameness result on invariant curves of near-parabolic maps: Let $g(z)=λz+O(z^2)$ have a non-degenerate parabolic fixed point at $0$ with multiplier $λ$ a primitive $q$th root of unity, and let $γ: \, ]-\infty,0] \to {\mathbb D}(0,r)$ be a $g^{\circ q}$-invariant curve landing at $0$ in the sense that $g^{\circ q}(γ(t))=γ(t+1)$ and $\lim_{t \to -\infty} γ(t)=0$. Take a sequence $g_n(z)=λ_n z+O(z^2)$ with $|λ_n|\neq 1$ such that $g_n \to g$ uniformly on ${\mathbb D}(0,r)$ and suppose each $g_n$ admits a $g_n^{\circ q}$-invariant curve $γ_n: \, ]-\infty,0] \to {\mathbb C}$ such that $γ_n \to γ$ uniformly on the fundamental segment $[-1,0]$. If $λ_n^q \to 1$ non-tangentially, then $γ_n$ lands at a repelling periodic point near $0$, and $γ_n \to γ$ uniformly on $]-\infty,0]$. In the special case of polynomial maps, this proves Hausdorff continuity of external rays of a given periodic angle when the associated multipliers approach a root of unity non-tangentially.

math.DS↗
arXiv2024

Value Distributions of Derivatives of $K$-regular Polynomial Families

Let $Ω\in \mathbb{C}$ be a domain such that $K:= \mathbb{C} \setminus Ω$ is compact and non-polar. Let $g_Ω$ be the Green's function with a logarithmic pole at infinity, and let $ω= ω_K$ be the equilibrium distribution on $K$. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty$, and let $(q_k^m)_k$ denote the sequence of $m$-th derivatives. We provide conditions, which ensure that the preimages $(q_k^m)^{-1}(\{a\})$ uniformly equidistribute on $\partial Ω$, as $k \to \infty$, for every $a \in \mathbb{C}$ and every $m = 0, 1, \ldots$

math.CV↗
arXiv2024

Convergence of Equilibrium Measures under $K$-regular Polynomial Sequences and their Derivatives

Let $K\subset\mathbb{C}$ be non-polar, compact and polynomially convex. We study the limits of equilibrium measures on preimages of compact sets, under $K$-regular sequences of polynomials, that center on $K$ and under the sequences of derivatives of all orders of such sequences. We show that under mild assumptions such limits always exist and equal the equilibrium measure on $K$. From this we derive convergence of the equilibrium distributions on the Julia sets of the sequence of polynomials and their derivatives of all orders.

math.DS↗
arXiv2024

The Parabolic Mandelbrot Set

We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from $M$ to $M_1$ is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal extension to complements.

math.DS↗
arXiv2024

Zero distributions of derivatives of polynomial families centering on a set

Suppose $C \subset \mathbb{C}$ is compact. Let $q_k$ be a sequence of polynomials of degree $n_k \to \infty$, such that the locus of roots of all the polynomials is bounded, and the number of roots of $q_k$ in any closed set $L$ not meeting $C$ is uniformly bounded. Supposing that $(q_k)_k$ has an asymptotic root distribution $μ$ we provide conditions on $C$ and $μ$ assuring the sequence of $m$th derivatives $(q_k^{(m)})_k$ also has asymptotic root distribution $μ$ for any $m\geq 1$. This complements recent results of Totik.

math.CV↗
arXiv2022

Holomorphic Explosions

This paper concerns the problem of extending the parameter domain of holomorphic motions to include isolated boundary points, punctures of the domains. Supposing that the parameter domain has an isolated boundary point $λ^*$, we explore the extension properties of the holomorphic motion to $λ^*$.

math.DS↗
arXiv2021

Conformal renormalization of compact sets

This paper develops a conformal renormalization scheme for compact sets $K \subset \mathbb{C}$. As one application of the conformal renormalization scheme we prove that for every isolated non-trivial connected component $E \subset K$ there exists a conformal homeomorphism $ϕ$ mapping a neighbourhood of $E$ into $\mathbb{C}$ such that the equilibrium measure on $K$ restricted to $E$ equals the scaled push-forward by $ϕ^{-1}$ of the equilibrium measure on $ϕ(E)$. Moreover the proof shows that the condition of connectedness of $E$ can be relaxed considerably. We also introduce an inverse to the procedure of conformal renormalization, which allows one to reconstruct $K$ from its conformal renormalizations.

math.CV↗
arXiv2021

Filled Julia sets of Chebyshev polynomials

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K in C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.

math.DS↗
arXiv2020

Conformal Equivalence of Measures and Dynamics of Orthogonal Polynomials

We introduce a notion of asymptotically orthonormal polynomials for a Borel measure $μ$ with compact nonpolar support in $\mathbb{C}$. Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and filled Julia sets to those for sequences of orthonormal polynomials. We give examples of measures for which the monic orthogonal polynomials are asymptotically orthonormal. Combining this with observations on conformal invariance of orthogonal polynomials we explore the measure dependency of the associated dynamics of orthogonal polynomials. Concretely, we study the dynamics of sequences of asymptotically orthonormal polynomials for the pullback measure $ϕ^\ast(μ)$ under affine mappings $ϕ$. We prove that the sequences of Julia sets and filled Julia sets of affine deformations of sequences of asymptotically orthonormal polynomials for $μ$ also have the same convergence properties as the Julia sets and filled Julia sets of the orthonormal polynomials. This leads to theorems on the convergence properties of affine deformations of the family of iterates of any fixed monic centered polynomial and, in the case the polynomial is hyperbolic, on the corresponding family of affine parameter spaces.

math.DS↗
arXiv2020

Weak limits of the measures of maximal entropy for Orthogonal polynomials

In this paper we study the sequence of orthonormal polynomials $\{P_n(μ; z)\}$ defined by a probability measure $μ$ with non-polar compact support $S(μ)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of measures of maximal entropy $ω_n$ for $P_n$ is contained in the polynomial-convex hull of $S(μ)$. And for $n$-th root regular measures the $ω_n$ converge weak* to the equilibrium measure on $S(μ)$.

math.DS↗
arXiv2017

Julia sets of Orthogonal polynomials

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials $\{P_n\}$ to properties of the support. More precisely we relate the Julia set of $P_n$ to the outer boundary of the support, the filled Julia set to the polynomial convex hull $K$ of the support, and the Green's function associated with $P_n$ to the Green's function for the complement of $K$.

math.CV↗
arXiv2016

On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

In the paper 'On the dynamics of polynomial-like mappings' Douady and Hubbard introduced the notion of polynomial-like maps. They used it to identify homeomophic copies of the Mandelbrot set inside the Mandelbrot set. They conjectured that in case of primitive copies the homeomorphism between the homeomorphic copy of the Mandelbrot set and the Mandelbrot set is q.-c., and similarly in the satellite case, it is q.-c. off any small neighborhood of the root. These conjectures are now Theorems due to Lyubich. The satellite copies of the Mandelbrot set are clearly not q-c homeomorphic to the Mandelbrot set. But are they mutually q-c homeomorphic? Or even q-c homeomorphic to half of the logistic Mandelbrot set? In this paper we prove that, in general, the induced Douady-Hubbard homeomorphism is not the restriction of a q-c homeomorphism: For any two satellite copies of the Mandelbrot set, the induced Douady-Hubbard homeomorphism is not q-c, if the root multipliers, which are primitive q and q' roots of unity, have q different from q'.

math.DS↗
arXiv2013

On The Notions of Mating

The different notions of matings of pairs of equal degree polynomials are introduced and are related to each other as well as known results on matings. The possible obstructions to matings are identified and related. Moreover the relations between the polynomials and their matings are discussed and proved. Finally holomorphic motion properties of slow-mating are proved.

math.DS↗