相关论文: Non-computable Julia sets
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 prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite)…
We show that there exist real quadratic maps of the interval whose attractors are computationally intractable. This is the first known class of such natural examples.
Recently, the place of the main programming language for scientific and engineering computations has been little by little taken by Julia. Some users want to work completely within the Julia framework as they work within the Python…
This article proposes an artistic approach to increase and enrich the understanding of Julia Sets. This approach includes the mathematical, the playful, the artistic and the computational dimensions. It is argued that these four dimensions…
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 propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum…
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…
We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.
Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers, and $\phi\in K(z)$ be a rational map of degree at least $2$. We prove that the $K$-Julia set of $\phi$ is the natural restriction of $\mathbb{C}_p$-Julia set,…
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 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…
This paper discusses "computational" systems capable of "computing" functions not computable by predefined Turing machines if the systems are not isolated from their environment. Roughly speaking, these systems can change their finite…
We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpi\'{n}ski curve
In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of…
A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this…
This paper is a reflexion on the computability of natural language semantics. It does not contain a new model or new results in the formal semantics of natural language: it is rather a computational analysis of the logical models and…
This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.
The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are…
We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic…