# Self Study of Fractals

I am looking for a book to self-study fractals with a certain criteria. I have checked out Getting Aquainted with Fractals. Note that Getting Aquainted with Fractals does not include exercises/problems. I would like to find a book that has exercises/problems, geometrical interpretations and more explanations (I do not like textbooks that are full of fluff, but I also do not think I am ready to fill in every blank either). To clarify about the last criteria, I do like books that are in the definition, theorem, proof format, however I would like a book that presents easier to read proofs that contain more detail. Also, I would like some discussions that prelude definitions which give their motivation and possibly give rise to geometric intuition.

I would also like a supplementary text covering the material in Mathematical Analysis as a lot of vocabulary, concepts and ideas come from analysis.

I know the basic concepts and properties: Self-Similarity, Recursively defined, and the Box Counting Dimension. I know about the classic fractals: Koch Curve, Sierpinski triangle and their variations. However, I do not know anything about fractals defined in the complex plane such as Mandelbrot set and Julia set. I also do not know much about the link between fractal geometry and chaotic theory.

Thanks for all the help!

My favourite book on fractals is Measure, Topology, and Fractal Geometry by Edgar. A short book and not very well known. It has a great many exercises all very suitable at undergrad. level but it requires a good mathematical background in basic analysis and topology. Apart from focusing on the geometrical aspects it also gives an excellent overview on different fractal dimensions definitions. But if I remember correctly this book does not cover fractals in the complex plane (Julia sets)

However, if you just want an overview on fractals I would start with Chaos and Fractals New Frontiers of Science by Peitgen, Jurgens & Saupe. It does not really give definitions but it guides the reader with simple examples or numerical experiments to the interesting properties of fractals.

A good understanding of Julia sets unfortunately requires a good understanding of Riemann surfaces, Hyperbolic geometry and Complex Analysis. The only book I know which also covers the necessary background is the Dynamics in one complex variable by (Fields medal winner) John Milnor. This is quite a tough book I read it when I was an undergrad. but I only fully understood it when I was a grad. student. But the material is great it covers Julia sets both from an analytical and topological view. It contains a collections of amazing proofs e.g. which Julia sets are smooth, when is a (polynomial) Julia set connected and a great many fun and insightful exercises.

I am actually not sure what level you are looking for but let me order the books based on difficulty (lowest easiest). Also perhaps you might be interested in reading Falconer's Fractal Geometry book. I only read some chapters many years ago, unfortunately the author introduces the Hausdorff dimension somewhere in the beginning which is perhaps one of the most difficult dimension definitions but after that it gets more easier.

1. Chaos and Fractals New Frontiers of Science by Peitgen. Jurgens & Saupe (No real proofs, but contains numerics and gives excellent intuition. Also the authors always put references in their informal statements to books/articles where you can find the details)

2. Fractal Geometry by Falconer (only the beginning is difficult, but contains mathematical rigour and simple proofs.)

3. Measure, Topology, and Fractal Geometry (nice exercises, requires good background in basic analysis and topology. Note it also contains a section on basic topology/metric spaces)

4. Dynamics in one complex variable by John Milnor (Only Julia sets, very difficult, but also very awesome)

• I'll definitely check it out! It sounds promising! – CodeKingPlusPlus Aug 5 '13 at 12:55
• Thanks so much! I'll probably need a supplementary text in analysis. Again I know the basics: $\epsilon-\delta$ definitions (which is a lot!) and basic point-set topology. Do any of these books link chaotic theory to fractals? I read online that fractal geometry is the language of chaos theory. Apparently, some fractals also are sensitive to initial conditions. – CodeKingPlusPlus Aug 5 '13 at 13:07
• I would start with Peitgen Jurgens Saupe the Bilbiography is a great many pages so if you want to know more about a given topic or want to know the details they always refer you to other books. – Novo Aug 5 '13 at 13:10
• +1: @CodeKingPlusPlus: Chaos and Fractals New Frontiers of Science by Peitgen, Jurgens & Saupe should be recommended to anyone learning the subject ! – Raymond Manzoni Aug 5 '13 at 13:25
• @CodeKingPlusPlus A word of caution though. The Peitgen, Jurgens & Saupe is almost a thousand pages and covers an immense amount of topics. Beaware not to drown in it ;) – Novo Aug 5 '13 at 13:31