Related papers: Capture components of cubic Siegel polynomials

A cubic polynomial $f$ with a periodic Siegel disk containing an eventual image of a critical point is said to be a \emph{Siegel capture polynomial}. If the Siegel disk is invariant, we call $f$ a \emph{IS-capture polynomial} (or just an…

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at…

Under conjugation by affine transformations, the dynamical moduli space of cubic polynomials $f$ with a $2$-cycle of Siegel disks is parameterized by a three-punctured complex plane as a degree-$2$ cover. Assuming the rotation number of…

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given…

Let $\alpha$ be an irrational number of sufficiently high type and suppose $P_\alpha(z)=e^{2\pi i\alpha}z+z^2$ has a Siegel disk $\Delta_\alpha$ centered at the origin. We prove that the boundary of $\Delta_\alpha$ is a Jordan curve, and…

We consider cubic polynomials f(z)=z^3+az+b defined over the function field C(L), with a marked point of period N and multiplier L. In the case N=1, there are infinitely many such objects, and in the case N>2, only finitely many. The case…

We prove that, for polynomials, the boundary of any bounded Fatou component is a Jordan curve, except maybe for Siegel disks.

We completely characterize the conformal radii of Siegel disks in the family $$P_\theta(z)=e^{2\pi i\theta}z+z^2,$$ corresponding to {\bf computable} parameters $\theta$. As a consequence, we constructively produce quadratic polynomials…

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot…

We prove that for typical rotation numbers polynomial Siegel disks are Jordan domains with boundaries containing at least one critical point.

We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist…

Let $f_\theta(z)=e^{2\pi i\theta}z+z^2$ be the quadratic polynomial having an indifferent fixed point at the origin. For any bounded type irrational number $\theta\in\mathbb{R}\setminus\mathbb{Q}$ and any rational number $\nu\in\mathbb{Q}$,…

Let $F$ be a non-archimedean local field of odd residual characteristic. We compute the Jordan set of a simple cuspidal representation of a symplectic group over $F$, using explicit computations of generators of the Hecke algebras of covers…

We prove: If $f(z)$ is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of $f$ is a Jordan curve. If $f(z)$ is a hyperbolic…

This article focus on the connected locus of the cubic polynomial slice $Per_1(\lambda)$ with a parabolic fixed point of multiplier $\lambda=e^{2\pi i\frac{p}{q}}$. We first show that any parabolic component, which is a parallel notion of…

We prove that the set of complex irrationals whose partial quotients in their Hurwitz continued fraction expansion are naturally regarded as subsets of $\mathbb Z^2$ and contain infinitely many homothetic copies of any finite subset of…

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the $main\, cubioid$ in…

In this paper we explore a class of quadratic polynomials having Siegel disks with unbounded type rotation numbers. We prove that any boundary point of Siegel disks of these polynomials is a Lebesgue density point of their filled-in Julia…

Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…