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Computability of Julia Sets (Algorithms and Computation in Mathematics), by Mark Braverman, Michael Yampolsky
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Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. This accessible book summarizes the present knowledge about the computational properties of Julia sets in a self-contained way.
- Sales Rank: #9482278 in Books
- Published on: 2010-11-22
- Released on: 2010-11-22
- Original language: English
- Number of items: 1
- Dimensions: 9.00" h x .38" w x 6.00" l, .54 pounds
- Binding: Paperback
- 151 pages
Review
From the reviews:
“The study of dynamical systems has at its core … a very computational feel. … One can feel the book trying to be self-contained … . The subject of the book is timely and important. … The questions posed and answered in the present book are natural and the approach well-suited to produce enlightening results. … The book is also generous … . It has the potential to inspire considerable future work in this intriguing field.” (Wesley Calvert, SIGACT News, Vol. 41 (1), 2010)
“Written in an accessible way with many explications, examples and illustrations. … this book sees the meeting of two worlds: computability theory and iteration of rational maps. It is a fruitful one … and a share of surprises. It is also a compendium of several years of research by the authors … together with a lot of new results. … a nice and quick introduction to both topics, and much of it is pleasant to read … . includes interesting discussions and presents stimulating conjectures.” (Arnaud Chéritat, Foundations of Computational Mathematics, Vol. 12, 2012)
From the Back Cover
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 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.
About the Author
M. Braverman is an expert in Theoretical Computer Science, particularly in applications of computability to Complex Analysis and Dynamical Systems
M. Yampolsky is an expert in Dynamical Systems, particularly in Holomorphic Dynamics and Renormalization Theory
Most helpful customer reviews
0 of 1 people found the following review helpful.
I should have checked it out of the library instead of buying it
By Roger Bagula
I bought this book because some programs for Julia sets on the internet
are under Mark Braverman's name.
I hate wasting money on books so I have been studying this one carefully.
These fellows mention Cremer points without ever talking about Schröder's equation.
If there was one piece of new material or anything that could be remotely considered original here,
I have sure missed it. I gave Carlson's Complex Dynamics (Universitext / Universitext: Tracts in Mathematics) a bad review for being very hard to read, so this less worthy effort
should get less as a book on computability that offers no programs or algorithms for such
that I can detect. Even the references seem to be sterile.
So buy Carlson's book or the Universal Mandelbrot Set: Beginning of the Story
,but check this one out from the library and don't buy it. I keep wondering how
these guys persuaded this company to publish this book?
...............................................................
I'm adding this after doing some web research: I was really curious about this book.
It seems the authors wrote a paper first about "uncomputable Julia sets": that is
when trying to compute z^2+c for a constant c the program on a Turing machine never stops.
So I naturally look under Chaitin in the index: just like
Schröder before, they have omitted him, even though they use
a 2^n "oracle" Turing machine. Essentially the argument seems to be that
for certain sets their is a never ending sequence that can't be found
as a limiting case. So that is why they got the book published.
Since most algorithms for computing Julia sets have Fatou plateaus
and it has been long known the Siegel disk plateau near the center of the Mandelbrot set
is a "pole", but that the pole Abs[f[z,n]-f[z,n-1]]->0. The values
of the plateau near the border set are what seems to be in question.
Still not worth the price of this book...
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