相关论文: The Boundedness Locus and baby Mandelbrot sets for…

For the family of complex rational functions of the form $R_{n,c,a}(z) = z^n + \dfrac{a}{z^n}+c$, known as ``Generalized McMullen maps'', for $a\neq 0$ and $n \geq 3$ fixed, we study the boundedness locus in some one-dimensional slices of…

First, for the family P_{n,c}(z) = z^n + c, we show that the geometric limit of the Mandelbrot sets M_n(P) as n tends to infinity exists and is the closed unit disk, and that the geometric limit of the Julia sets J(P_{n,c}) as n tends to…

The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic…

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 $M$…

We study the mean-median map as a dynamical system on the space of finite sets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive…

We study the dynamics of template iterations, consisting of arbitrary compositions of functions chosen from a finite set of polynomials. In particular, we focus on templates using complex unicritical maps in the family $\{ z^d + c, c \in…

We study a poset of compositions restricted by part size under a partial ordering introduced by Bj\"{o}rner and Stanley. We show that our composition poset $C_{d+1}$ is isomorphic to the poset of words $A_d^*$. This allows us to use…

For the family of complex rational functions of the form R(z)= z^n + a/z^n+b, known as "Generalized McMullen maps", for non-zero a, and integer n fixed and at least 3, we describe the apparent phenomena of baby Julia sets in parameter space…

We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the…

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near…

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the…

We prove that given any $\theta_1,\ldots,\theta_{2d-2}\in \R\setminus\Z$, the support of the bifurcation measure of the moduli space of degree $d$ rational maps coincides with the closure of classes of maps having $2d-2$ neutral cycles of…

For the family of complex rational functions known as "Generalized McMullen maps", F(z) = z^n + a/z^n+b, for complex parameters a and b, with a nonzero, and any integer n at least 3 fixed, we reveal, and provide a combinatorial model for,…

Let $\langle K,\nu \rangle$ be a real closed valued field, and let $S\subseteq K^n$ be an open semi-algebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on $S$, only values…

The set \[ \overline{\mathbb{E}}= \{ x \in {\mathbb{C}}^3: \quad 1-x_1 z - x_2 w + x_3 zw \neq 0 \mbox{ whenever } |z| < 1, |w| < 1 \} \] is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex,…

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof…

A topological ring R, an escape set B in R and a family of maps z^d+c defines the degree d Mandelstuff as the set of parameters for which the closure of the orbit of 0 does not intersect R. If B is the complement of a ball of radius 2 in C…

Consider the one-parameter family of cubic polynomials defined by $f_t(z) =-\frac 32 t(-2z^3+3z^2)+1, t \in \mathbb{C}_2$. This family corresponds to a slice of the parameter space of cubic polynomials in $\mathbb{C}_2[z]$. We investigate…

Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the…

In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and…