Quadratic Family Research Articles

From hyperbolic to parabolic parameters along internal rays

For the quadratic family f c ( z ) = z 2 + c f_{c}(z) = z^2+c with c c in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter c c converges to a parabolic parameter c ^ {\hat {c}} radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on c ^ {\hat {c}} . We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at c ^ {\hat {c}} with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].

Preperiodic points with local rationality conditions in the quadratic unicritical family

Preperiodic points with local rationality conditions in the quadratic unicritical family

A Spectral Method Approach to Quadratic Normal Volatility Diffusions

This paper is concerned with the quadratic volatility family of driftless stochastic differential equations (SDEs), also known in the literature as Quadratic Normal Volatility models (QNV), which have found applications primarily in mathematical finance, but can also model dynamics of stochastic processes in other fields such as mathematical biology and physics. These SDE models are characterized by a quadratic volatility term and can be reduced to one of four distinct possibilities depending on the roots of the quadratic volatility term and the position of initial value. We find explicit weak solutions for each case by a combination of Itô calculus and Fourier analysis, which can be described as a ’spectral method’. Furthermore, for all cases we also express the weak solutions as fairly simple functions of Brownian motion, which allows for efficient one-step Monte Carlo evaluation of functionals of the solutions Xt of the form E(f(Xt)) and also more general functionals of the solution. The method used to compute the solutions may also be of interest itself more generally in other fields where SDEs play a fundamental role.

Splitting and coexistence of 2-D strange attractors in a general family of Expanding Baker Maps

We consider a two-parameter family of Expanding Baker Maps on the plane, being a > 1 and an expansion rate and a rotation angle, respectively. We prove that exhibits strange attractors for every a sufficiently close to 1. We also study how such attractors may split into other ones of a larger number of connected pieces as a decreases to 1 and is a rational number. The study of the family is strongly motivated by the rich dynamics observed for the quadratic family .

On the Average of p-Selmer Ranks in Quadratic Twist Families of Elliptic Curves Over Global Function Fields

Abstract Let $\mathbb{F}_{q}$ be a finite field whose characteristic is relatively prime to $2$ and $3$. Let $p$ be a prime number that is coprime to $q$. Let $E$ be an elliptic curve over the global function field $K = \mathbb{F}_{q}(t)$ such that $\textrm{Gal}(K(E[p])/K)$ contains the special linear group $\textrm{SL}_{2}(\mathbb{F}_{p})$. We show that if the quadratic twist family of $E$ has an element whose Néron model has a multiplicative reduction away from $\infty $, then the average $p$-Selmer rank is $p+1$ in large $q$-limit for almost all primes $p$.

Critical recurrence in the real quadratic family

AbstractWe study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate $(\delta _n)$ of the critical orbit such that, for almost every non-regular parameter a, the set of n such that $\vert F^n(0;a) \vert < \delta _n$ is infinite. In particular, when $\delta _n = n^{-1}$ , this extends an earlier result by Avila and Moreira [Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2)161(2) (2005), 831–881].

Low-lying zeros in families of elliptic curve L-functions over function fields

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.

Rank jumps on elliptic surfaces and the Hilbert property

Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.

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

AbstractA 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 ${\mathcal X}$ .

Lebesgue measure of Feigenbaum Julia sets

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ the corresponding set of parameters $c$ is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set.

Simultaneous Bifurcation of Limit Cycles and Critical Periods

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

Tiling number problem and $\boldsymbol{\theta}$-congruent number problem

The tiling number problem is closely related to the arithmetic of elliptic curves. In this paper, we construct a family of tiling numbers with multiple prime factors by using Birch lemma and the induction method developed by one of the authors(Ye Tian) to prove the non-torsioness of Heegner points. In addition, we prove the $2$-part BSD conjecture of the related elliptic curves. We deal with the quadratic twist family of elliptic curves without complex multiplication and the distribution of $2$-Selmer groups of this family is not predicted by the general conjecture or the known results.

Quadrature domains and the real quadratic family

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup ofPSL(2,Z)\mathrm {PSL}(2,\mathbb {Z}), and that this conformal mating is the Schwarz function of a simply connected quadrature domain.

Total positivity in exponential families with application to binary variables

We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for existence of the MLE. Quadratic exponential familes of MTP2 distributions contain attractive Gaussian graphical models and ferromagnetic Ising models as special examples. We show that these are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and leads to sparsity in the estimated graphical model. We prove that the maximum likelihood estimator (MLE) in an MTP2 binary exponential family exists if and only if both of the sign patterns $(1,-1)$ and $(-1,1)$ are represented in the sample for every pair of variables; in particular, this implies that the MLE may exist with $n=d$ observations, in stark contrast to unrestricted binary exponential families where $2^d$ observations are required. Finally, we provide a novel and globally convergent algorithm for computing the MLE for MTP2 Ising models similar to iterative proportional scaling and apply it to the analysis of data from two psychological disorders.

Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modular curve $X_0(3p)$ has infinitely many $F$-rational points the method applies to ``most'' elliptic curves $E$ having a cyclic $3p$-isogeny. It also applies in certain cases when $X_0(3p)$ has only finitely many points. For example, we find an elliptic curve over $\mathbb{Q}$ for which a positive proportion of quadratic twists have an element of order $5$ in their Tate-Shafarevich groups. The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime $p \equiv 1 \pmod 9$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate-Shafarevich groups.

On the maximal invariant set for the map x2 – 2 restricted to intervals

In this paper, we study the maximal invariant set of a quadratic family related to a class of unimodal maps. This family is very important and have direct application in many branches of science. In particular, we characterize when the maximal invariant of f(x) = x2 − 2 (restricted to an interval) has a chaotic behavior.

Fractional Susceptibility Functions for the Quadratic Family: Misiurewicz–Thurston Parameters

For the quadratic family, we define the two-variable ($\eta$ and $z$) fractional susceptibility function associated to a C^1 observable at a stochastic map. We also define an approximate, "frozen" fractional susceptibility function. If the parameter is Misiurewicz-Thurston, we show that the frozen susceptibility function has a pole at $z=1$ for generic observables if a "one-half" transversality condition holds. We introduce "Whitney" fractional integrals and derivatives on suitable sets $\Omega$. We formulate conjectures supported by our results on the frozen susceptibility function and numerical experiments. In particular, we expect that the fractional susceptibility function for $\eta=1/2$ is singular at $z=1$ for Collet-Eckmann maps and generic observables. We view this work as a step towards the resolution of the paradox that the classical susceptibility function is holomorphic at $z=1$ for Misiurewicz-Thurston maps, despite lack of linear response.

Equidistribution of critical points of the multipliers in the quadratic family

A parameter c0∈C in the family of quadratic polynomials fc(z)=z2+c is a critical point of a period n multiplier, if the map fc0 has a periodic orbit of period n, whose multiplier, viewed as a locally analytic function of c, has a vanishing derivative at c=c0. We prove that all critical points of period n multipliers equidistribute on the boundary of the Mandelbrot set, as n→∞.

Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters

In this work the homoclinic bifurcation of the family H={h(a,b)(x)=ax2+b:a∈R/{0},b∈R} is studied. We proved that this family has a homoclinic tangency associated to x=0 of P1 for b=-2/a. Also we proved that Wu(P1) does not intersect the backward orbit of P1 for b>-2/a, but has intersection for b0. So H has this type of the bifurcation.

On fair entropy of the tent family

<p style='text-indent:20px;'>The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [<xref ref-type="bibr" rid="b13">13</xref>] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy <inline-formula><tex-math id="M1">\begin{document}$ h(a) $\end{document}</tex-math></inline-formula> of the tent map <inline-formula><tex-math id="M2">\begin{document}$ f_a $\end{document}</tex-math></inline-formula>, as a function of the parameter <inline-formula><tex-math id="M3">\begin{document}$ a = \exp(h_{top}(f_a)) $\end{document}</tex-math></inline-formula>, is continuous and strictly increasing on <inline-formula><tex-math id="M4">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula>. In this short note, we extend the last result and characterize regularity of the function <inline-formula><tex-math id="M5">\begin{document}$ h $\end{document}</tex-math></inline-formula> precisely. We prove that <inline-formula><tex-math id="M6">\begin{document}$ h $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula>-Hölder continuous on <inline-formula><tex-math id="M8">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula> and identify its best Hölder exponent on each subinterval of <inline-formula><tex-math id="M9">\begin{document}$ [\sqrt{2},2] $\end{document}</tex-math></inline-formula>. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [<xref ref-type="bibr" rid="b7">7</xref>], we give a formula of pointwise Hölder exponents of <inline-formula><tex-math id="M10">\begin{document}$ h $\end{document}</tex-math></inline-formula> at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of <inline-formula><tex-math id="M11">\begin{document}$ h $\end{document}</tex-math></inline-formula> vanishes almost everywhere.