Searcharxiv⌕ Search

arXiv subjects

Eva Uhre

Publications and source records attributed to Eva Uhre.

8 recordsLinked to original sources
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 $\mu_k$ of $q_k$ converge weak* to some compactly supported measure $\mu$, 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 $p_\mu$ 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

Viscous liquid dynamics modeled as random walks within overlapping hyperspheres

The hypersphere model is a simple one-parameter model of the potential energy landscape of viscous liquids, which is defined as a percolating system of same-radius hyperspheres randomly distributed in $\mathbb{R}^{3N}$ in which $N$ is the number of particles. We study random walks within overlapping hyperspheres in 12 to 45 dimensions, i.e., above the percolation threshold, utilizing an algorithm for on-the-fly placement of the hyperspheres in conjunction with the kinetic Monte Carlo method. We find behavior typical of viscous liquids; thus decreasing the hypersphere density (corresponding to decreasing the temperature) leads to a slowing down of the dynamics by many orders of magnitude. The shape of the mean-square displacement as a function of time is found to be similar to that of the Kob-Andersen binary Lennard-Jones mixture and the Random Barrier Model, which predicts well the frequency-dependent fluidity of nine glass-forming liquids of different chemistry [Bierwirth et al., Phys. Rev. Lett. $\mathbf{119}, 248001\,(2017)$].

cond-mat.soft↗
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↗
arXiv2023

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↗
arXiv2023

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

Let $\Omega \in \mathbb{C}$ be a domain such that $K:= \mathbb{C} \setminus \Omega$ is compact and non-polar. Let $g_\Omega$ be the Green's function with a logarithmic pole at infinity, and let $\omega = \omega_K$ 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 $\partial \Omega$, as $k \to \infty$, for every $a \in \mathbb{C}$ and every $m = 0, 1, \ldots$

math.CV↗
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

Limits of Quadratic Rational Maps: The Cantor Locus

The \emph{Cantor locus} is the unique hyperbolic component, in the moduli space of quadratic rational maps ${\bf rat}_2$, consisting of maps with totally disconnected Julia sets. Whereas the geometry and dynamics of the Cantor locus is well understood, its boundary and the dynamics of the maps on the boundary are not. In this paper, we explore the dynamics near the parabolic parts of the boundary. We introduce the concept \emph{dynamical marking} of a map $g$, relative to the quadratic, parabolic polynomial $\mathrm{P}_{\opq}(z)={\opq} z+z^2$, with $\opq=e^{2\pi ip/q}$. A dynamical marking $(x,\psi)$ of $g$ is a conjugacy $\psi$ between $\mathrm{P}_{\opq}$ (on its parabolic basin of 0) and $g$, which \emph{marks} the dynamical position of the critical values $v_1=\psi(-\frac{\lambda^2}{4})$, $v_2=\psi(x)$ of $g$. We construct a local parametrization of the Cantor locus, which parametrizes by dynamical marking, and use it to prove a form of \emph{stability} of dynamical marking. That is, for sequences in the Cantor locus, of fixed dynamical marking $x$ and such that the eigenvalue $\lambda_k$ of the attracting fixed point tends to $\opq$ \emph{subhorocyclicly}, either the sequence converges to the unique parabolic parameter in the boundary, which has a fixed point eigenvalue $\opq$ and which is marked by $x$ relative to $\mathrm{P}_{\opq}$. Or, the sequence tends to infinity in ${\bf rat}_2$, and certain representatives $G_{\lambda_k,a_k}$ have \emph{rescaled} limits in the boundary of the Cantor locus within ${\bf rat}_2$.

math.DS↗