Related papers: Constructing Non-Computable Julia Sets

We partially answer to a question of Vidaux and Videla by constructing an infinite family of rings of algebraic integers of totally real subfields of Q whose Julia Robinson's Number is distinct from 4 and +$\infty$. Moreover the set of the…

We prove the existence of rational maps having smooth degenerate Herman rings. This answers a question of Eremenko affirmatively. The proof is based on the construction of smooth Siegel disks by Avila, Buff and Ch\'{e}ritat as well as the…

In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its…

We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that…

We analyse the SU(2)_k WZNW models beyond the integrable representations and in particular the case of SU(2)_0. We find that these are good examples of logarithmic conformal field theories as indecomposable representations are naturally…

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…

In this paper, we consider the problem of determining the \emph{exact} number of periodic orbits for polynomial planar flows. This problem is a variant of Hilbert's 16th problem. Using a natural definition of computability, we show that the…

We extend results about the dimension of the radial Julia set of certain exponential functions to quasiregular Zorich maps in higher dimensions. Our results improve on previous estimates of the dimension also in the special case of…

Let $S$ be a unital ring, $S[t;\sigma,\delta]$ a skew polynomial ring where $\sigma$ is an injective endomorphism and $\delta$ a left $\sigma$-derivation, and suppose $f\in S[t;\sigma,\delta]$ has degree $m$ and an invertible leading…

In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in \cite{Bou59} and a result in \cite{PU07} to obtain two non-Abelian indecomposable solvable…

The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for $g\leq 3$ and the surjectivity of the theta map follows. Instead it is known that this map is not…

Let $p\in \mathbb{N}$ be prime, and let $\theta$ be irrational. The authors have previously shown that the noncommutative $p$-solenoid corresponding to the multiplier of the group $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$…

We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $\Psi $, a group cocycle for $GL_{n} (\Z )$; the special values are computed as…

Using the skew-symmetry of the differential operators and multiplication operators in the canonical representations of finite-dimensional classical Lie algebras, we obtain some noncanonical polynomial representations of the classical Lie…

We produce a family of numerical Campedelli surfaces with \Z/6 torsion by constructing the (Gorenstein codimension 5) canonical ring of the \'{e}tale six to one cover using serial unprojection. In Section 2 we develop the necessary…

We study the cyclic $U(\mathfrak{gl}_n)$-module generated by the $l$-th power of the $\alpha$-determinant. When $l$ is a non-negative integer, for all but finite exceptional values of $alpha$, one shows that this cyclic module is isomorphic…

We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to $\psi^\bullet$-quadratic-like Siegel maps. In this form, the bounds are compatible with the $\psi$-quadratic-like renormalization theory and are easily transferable…

We introduce a notion of asymptotically orthonormal polynomials for a Borel measure $\mu$ with compact nonpolar support in $\mathbb{C}$. Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and…

We construct holomorphic maps with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a…

This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of…