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What are some examples of cutting-edge research involving fractals or self-similar structures?

Who's actively contributing high-quality research in this field?

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Your question seems a little ambiguous. Self-similarity is a very general phenomena and it happens in many places, not just with objects that are "geometric". You can have self-similarity of algebras and a variety of other kinds of objects. In that sense self-similarity isn't so much of an object of study in its own right, just a general phenomenon. The theorems about the Mandelbrot set that I know of are due to people like Fatou, Julia, Douady and Hubbard, which in some sense are before and after Mandelbrot, respectively. Could you be a little more clear about which field "this field" is? –  Ryan Budney Sep 26 '10 at 22:46
    
When I look at what Mandelbrot has done it seems his work is mostly recognised as physics, not mathematics. In that he popularized the notion of fractals as being relevant to physics, and physical sciences. –  Ryan Budney Sep 26 '10 at 23:06

4 Answers 4

Fractal antennae lend theory to the design of fractal solar panels.

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Behold the mandelbulb http://www.skytopia.com/project/fractal/mandelbulb.html !!!!

oddly enough the mandelbulb looks almost organic.

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render 3D fractals based on the original 2D mandelbrot formula, such as mandelbox and mandelbulb + many more - using free open source software called "mandelbulber"

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no mathematical interest whatsoever –  Glougloubarbaki Apr 3 '12 at 20:01

As pointed out in the comments, this is a tremendously broad area and there are many people doing a lot of very good research, so I can only say a little bit about the small part of it that I'm familiar with. When dealing with "chaotic" dynamical systems (a notion which can be made precise in a few different ways), one often finds that the chaotic behaviour of the dynamics is intimately connected to the presence of various geometric structures in the phase space that are best characterised as fractals -- the Julia sets found in complex dynamics are examples of this.

To study these structures, one uses various dimensional quantities, such as Hausdorff dimension. It turns out that some fundamental dynamical quantities such as topological entropy can also be defined as "dimensions" of a sort, so that there are deep connections between fractal geometry and chaotic dynamics. One of the standard (advanced) references on this is "Dimension Theory in Dynamical Systems", by Yakov Pesin. A more introductory exposition (with apologies for self-advertising) is "Lectures on Fractal Geometry and Dynamical Systems", by Pesin and Climenhaga.

As an example of the sort of thing that occurs, suppose you have a dynamical system with phase space $X$ and an observable function $\phi\colon X\to \mathbb{R}$. You take measurements of $\phi$ as time goes along, and calculate its average value as time goes to infinity. This average value is a function of your starting position $x\in X$. Let $K_\alpha$ denote the set of points for which that average value is equal to $\alpha$; then typically speaking, there is a range of values of $\alpha$ for which $K_\alpha$ is a fractal with quite an intricate structure. This is called a multifractal decomposition. One can define a multifractal spectrum by $B(\alpha) = \dim_H(K_\alpha)$, and it turns out (somewhat miraculously) that the function $\alpha\mapsto B(\alpha)$ is actually concave and analytic in a great many (important) examples! This multifractal analysis has deep connections to thermodynamic formalism and statistical properties of chaotic dynamical systems; a good reference is "Thermodynamics of Chaotic Systems", by Beck and Schlögl.

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