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As others have had great success with their question, I hope to ask one in a similar vein. As a student who has some background in complex analysis and dynamical systems, I am hoping to explore questions regarding connectivity (both connected and locally connected) in the Mandelbrot set. However, I have very little background in topology so I was wondering if some one could make recommendations of how to approach studying these questions.

I suppose I am really hoping for a focused introduction to connected in topology with a focus on complex dynamics.


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The Orsay notes have already been translated, but you want other references.

First read Milnor notes on external rays and local connectivity:

J. Milnor, "Periodic orbits, externals rays and the Mandelbrot set: an expository account," in Géométrie complexe et systèmes dynamiques: Colloque en l’honneur d’Adrien Douady, Paris: Soc. Mat. de France, 2000, pp. 277-333.

J. Milnor, "Local connectivity of Julia sets: expository lectures," in The Mandelbrot Set, Theme and Variations, Lei, T., Ed., Cambridge: Cambridge Univ. Press, 2000, pp. 67-116.

External rays are used to construct puzzle pieces. The pieces around the critical point 0 are nested (each one strictly containing the next. The goal is to compute the annuli between consecutive pieces and hopefully show their sum diverges. If so, the diameter of the pieces around 0 will shrink, and this gives you a system of neighborhoods that prove local connectivity of the Julia set at 0. A related construction gives the same result on the Mandelbrot set M. The details are in Hubbard's explanation of Yoccoz's proof of MLC for finitely renormalizable quadratic polynomials:

J. H. Hubbard, "Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz," in Topological Methods in Modern Mathematics, Houston, TX: Publish or Perish, 1993, pp. 467-511.

After understanding the mechanics of the puzzle construction, learn about Lyubich's principal nest. This is a subset of puzzle pieces with more complicated combinatorics:

M. Lyubich, "Dynamics of quadratic polynomials. I, II," Acta Math., vol. 178, iss. 2, pp. 185-247, 247, 1997.

This allows Lyubich to prove local connectivity of M at infinitely renormalizable pararameters of bounded type:

M. Lyubich, "Dynamics of quadratic polynomials. III: parapuzzle and SBR measures," Asterisque volume in honor of Adrien Douady's 60th birthday ``G\'eom\'etrie complexe et syst\'emes dynamiques'', v. 261 (2000), 173 - 200.

Then you are ready to read the most recent developments:

J. Kahn and M. Lyubich, "The quasi-additivity law in conformal geometry," Ann. of Math., vol. 169, iss. 2, pp. 561-593, 2009.

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Thanks for the Douady-Hubbard translation! I hope it will be published as a book sometime. –  Alexandre Eremenko Jan 6 '13 at 6:56
@Alexandre: Welcome. I just happened to find it last week. And YES... I would buy the book. –  Rodrigo A. Pérez Jan 6 '13 at 7:59
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As an introduction to complex dynamics (as well as connectivity and local connectivity) I strongly recommend the book of J. Milnor. On my opinion, this is by far the best book on the subject. After that, you want to know more about the Mandelbrot set, and for this I recommend the 2-volume preprint of Douady and Hubbard in Publ. Orsay. (Now available on Internet. But exists only in French). Then (or simultaneously) you read McMullen's book Complex dynamics and renormalization. Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ, 1994.

And then, papers of Yoccoz, Lyubich and others.

I have to warn you that this is a highly technical subject, and it will be hard to start your own research in it without a proper adviser.

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