Related papers: On uniformly disconnected Julia sets

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$)…

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperK\"ahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the…

In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a…

Quantum states that remain separable (i.e., not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely…

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…

A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane $\mathbb{R}^2$. Recognizing them is known to be $\exists\mathbb{R}$-complete, i.e., as hard as solving a system of polynomial…

We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the…

We find that characteristics of quantum tunneling in the presence of chaos can be regarded as a manifestation of the Julia set of the complex dynamical system. Several numerical evidences for the standard map together with a rigorous…

In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a…

Ahlfors and Gehring asked for the Riemann Mapping Theorem for quasiconformal mappings (QC) of R^3. We summarise our solution: (a) QC reflections are tame (b) T is the fixed set of a QC reflection iff T is a uniform sphere (i.e. the limits…

We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M; \mathbb{R})$ of $M$ is…

We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou…

Let $\mathcal{U}=\left[ \begin{array}{cc} \mathcal{A} & \mathcal{M} \mathcal{N}& \mathcal{B} \end{array} \right]$ be a generalized matrix ring, where $\mathcal{A}$ and $\mathcal{B}$ are 2-torsion free. We prove that if $\phi…

In this article we expound a discovery of the quantum equivalence/duality of U(N) noncommutative quantum field theories (NC QFT) related by the theta-exact Seiberg-Witten (SW) maps and at all orders in the perturbation theory with respect…

Let G be the group of rational points of a connected reductive group over a finite field. Based on work of Lusztig and Yun, we make the Jordan decomposition for irreducible G-representations canonical. It comes in the form of an equivalence…

We characterize those algebras over a disconnected uniformly complete topological field which are representable as algebras of continuous functions on compact topological spaces, generalizing thus Gelfand duality for non-archimedean normed…

This paper is devoted to the study of conformal maps of the unit disk $\mathbb{D}$ in the plane onto a bounded Jordan domain $G$. The main aim is to show that such a map is asymptotically symmetric if and only if $G$ is bounded by a…

Let M be a compact Kaehler manifold equipped with a Hamiltonian action of a compact Lie group G. In [Invent. Math. 67 (1982), no.~3, 515--538], Guillemin and Sternberg showed that there is a geometrically natural isomorphism between the…

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…