Related papers: Primitive tuning for non-hyperbolic polynomials

We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the…

Let $z\ne \pm1,w^2$ be a fixed integer, and let $f(t)\ne g(t)^2$ be a fixed polynomial over the integers. It is shown that the subset of primes $p\geq 2$ such that $z$ and $f(z)$ is a pair of simultaneous primitive roots modulo $p$ has…

Let $f: S^1\to S^1$ be a $C^3$ homeomorphism without periodic points having a finite number of critical points of power-law type. In this paper we establish real a-priori bounds, on the geometry of orbits of $f$, which are beau in the sense…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We…

We give a combinatorial definition of "core entropy" for quadratic polynomials as the growth exponent of the number of certain precritical points in the Julia set (those that separate the $\alpha$ fixed point from its negative). This notion…

Let X be a smooth curve defined over the algebraic numbers, let a,b be algebraic numbers, and let f_l(x) be an algebraic family of rational maps indexed by all l in X. We study whether there exist infinitely many l in X such that both a and…

A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are…

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric…

By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a…

In this paper we study a one parameter family of rational maps obtained by applying the Chebyshev-Halley root finding algorithms. We show that the dynamics near parameters where the family presents some degeneracy might be understood from…

The multiplier $\lambda_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical…

We show that for all homogeneous polynomials $ f_{m}$ of degree $m$, in $d$ variables, and each $j = 1, \dots , d$, we have \begin{equation*} \left\langle x_{j}^{2}f_{m},f_{m}\right\rangle _{L^{2}\left( \mathbb{S}% ^{d-1}\right) } \geq…

We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of…

We continue the study of straightening maps for the family of polynomials of degree $d \ge 3$. The notion of straightening map is originally introduced by Douady and Hubbard to study relationship between polynomial-like renormalizations and…

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There…

Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, the only…

We propose in this paper New Q-Newton's method. The update rule is very simple conceptually, for example $x_{n+1}=x_n-w_n$ where $w_n=pr_{A_n,+}(v_n)-pr_{A_n,-}(v_n)$, with $A_n=\nabla ^2f(x_n)+\delta _n||\nabla f(x_n)||^2.Id$ and…

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is…

We will show the Mandelbrot set $M$ is locally conformally inhomogeneous: the only conformal map $f$ defined in an open set $U$ intersecting $\partial M$ and satisfying $f(U\cap\partial M)\subset \partial M$ is the identity map. The proof…

In this paper, we consider a one-parameter family of degree $d\ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. We construct a polynomial whose roots correspond to parameter values for which the…