Computability Of Julia Sets
Download Computability Of Julia Sets PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get Computability Of Julia Sets book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages.
Computability of Julia Sets
Author: Mark Braverman
language: en
Publisher: Springer Science & Business Media
Release Date: 2009-02-08
Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content. Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computer screen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking - it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge (most of it from the authors' own work) about the computational properties of Julia sets in a self-contained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems.
Turing's Legacy
Author: Rod Downey
language: en
Publisher: Cambridge University Press
Release Date: 2014-05
A collection of essays celebrating the influence of Alan Turing's work in logic, computer science and related areas.
Computability and Complexity of Julia Sets
The field of study in the thesis is Real Computation, and Computability and Complexity in Dynamical Systems. The thesis consists of two parts. The first part is devoted to results on general computation of functions and sets over the reals, while in the second part results about computability and complexity in Complex Dynamics are presented. Studying computability and complexity over the reals is important for understanding the relationship between nature and computing, and for providing theoretical backing to fundamental problems in Numerical Analysis. In the first part of the thesis results unifying two most commonly used models of computation are presented. One is the model of Computable Analysis that is based on rational approximations of continuous objects, such as real functions and sets [Grz55, Ko91, Wei00]. The other is the Blum-Shub-Smale (BSS) model that is based on precise algebraic operations [BCSS98]. Insights obtained through this connection are then used to extend the computational complexity notion to some discontinuous functions. In the second part of the thesis, results on the computability and complexity of Julia sets are presented. Julia sets arise in one-dimensional complex dynamics. They have been intensely studied in the past 100 years, and since the 1980s numerous programs have been written to produce images of the sets, both for research and for their aesthetic value. We have studied computational properties of Julia sets, obtaining a virtually complete classification of the computational properties of Julia sets as well as complexity results, that complement the previous empirical work on producing their images.