Related papers: Computable geometric complex analysis and complex …
We discuss computability of impressions of prime ends of compact sets. In particular, we construct quadratic Julia sets which possess explicitly described non-computable impressions.
It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has…
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…
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.
In this paper we present an introduction to the area of computability in dynamical systems. This is a fairly new field which has received quite some attention in recent years. One of the central questions in this area is if relevant…
Frames play an important role in various practical problems related to signal and image processing. In this paper, we define computable frames in computable Hilbert spaces and obtain computable versions of some of their characterizations.…
We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of…
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…
In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of boundary is given either as a list of…
We study the complexity of deterministic and probabilistic inversions of partial computable functions on the reals.
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with…
We show that there exist real parameters $c$ for which the Julia set $J_c$ of the quadratic map $z^2+c$ has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold $T(n)$, there exist a…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…
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…
We discuss a classical complexity of finite-dimensional unitary transformations, which can been seen as a computable approximation of classical descriptional complexity of a unitary transformation acting on a set of qubits.
In this article, we provide the first theoretical framework guaranteeing that computers can, in principle, be used to analyze the parameter space of complex H\'{e}maps. More precisely, we obtain computability results for hyperbolic…
We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative $C^*$ algebras.