相关论文: Constructing Non-Computable Julia Sets
We prove that several dynamically defined fractals in $\mathbb{C}$ and $\mathbb{C}^2$ which arise from different type of polynomial dynamical systems can not be the same objects. One of our main results is that the closure of Misiurewicz…
We establish universality of the fat Basilica Julia set $J(z^2-\frac34)$ in conformal dynamics in the following sense: $J(z^2-\frac34)$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial as well as to the limit…
We prove that for typical rotation numbers polynomial Siegel disks are Jordan domains with boundaries containing at least one critical point.
A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are…
Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb…
In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof…
We consider the symmetries of Julia sets of polynomial skew products on C^2, which are birationally conjugate to rotational products. Our main results give the classification of the polynomial skew products whose Julia sets have infinitely…
In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels…
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire…
We compute a complete set of isomorphism classes of cubic fourfolds over $\mathbb{F}_2$. Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all…
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod $p$ by using theta series attached to positive definite quadratic forms.
Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational…
We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $\deg p_1\geq 1$, $\deg p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{\deg…
The dynamics of all quadratic Newton maps of rational functions are completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three…
In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an…
We investigate Siegel theta series for quadratic forms of signature $(m-1,1)$. On the one hand, we construct a holomorphic series that does not transform like a modular form. On the other hand, we construct a non-holomorphic series that…
If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic…
We study unilateral series in a single variable $q$ where its exponent is an unbounded increasing function, and the coefficients are periodic. Such series converge inside the unit disk. Quadratic polynomials in the exponent correspond to…