Related papers: Jordan mating is always possible for polynomials

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive…

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic…

Let $W$ be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that $W$ is birational to a product of a smooth projective variety $A$ and the projective line. We prove that if $A$ contains no rational…

In this paper we consider in detail the composition of an irreducible polynomial with X^2 and suggest a recurrent construction of irreducible polynomials of fixed degree over finite fields of odd characteristics. More precisely, given an…

We prove that every multiplicative bijective map, Jordan bijective map, and Jordan triple bijective map from a triangular algebra onto any ring is automatically additive.

A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…

We construct via Fra\"iss\'e amalgamation an $\omega$-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a `limit of $D$-relations'. The construction is based on a…

In the present paper we prove that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation. We also apply our technique to various Jordan algebras of infinite dimensional…

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x,y) in K[x,y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f+c) has at most two rational elements. When…

For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a…

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…

In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…

It is proved that the Chebyshev's method applied to an entire function $f$ is a rational map if and only if $f(z) = p(z) e^{q(z)}$, for some polynomials $p$ and $q$. These are referred to as rational Chebyshev maps, and their fixed points…

We prove the dynamical Manin-Mumford conjecture for regular polynomial maps of A^2 and irreducible curves avoiding super-attracting orbits at infinity, over any field of characteristic 0.

We construct a Jordan curve $\Gamma \subset \mathbb{C}$ so that for any rectifiable arc $\sigma$ with endpoints in distinct complementary components of $\Gamma$, $H^1(\sigma \cap \Gamma) > 0$.

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

Let $\MP_d$ denote the space of polynomials $f: \C \to \C$ of degree $d\geq 2$, modulo conjugation by $\Aut(\C)$. Using properties of polynomial trees (as introduced in [DM, math.DS/0608759]), we show that if $f_n$ is a divergent sequence…

Let $\mathcal{N}$ be a non-trivial and complete nest on a Hilbert space $H$. Suppose $d=\{d_n: n\in N\}$ is a group of linear mappings from Alg$\mathcal{N}$ into itself. We say that $d=\{d_n: n\in N\}$ is a Jordan higher derivable mapping…

Let $K$ be a field (finite or infinite) of char$(K)\neq 2$ and let $UT_n=UT_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot $ is the usual product on $UT_n$ then with the new product $a\circ b=(1/2)(a\cdot b…

We consider polynomial maps, which we call degree $d$-linear maps, that satisfy the Jacobian condition. We prove that certain infinite families of elements, which appear in the coefficients of the formal inverse of such maps, are in the…