I was researching fractals for my senior mathematics presentation and discovered that one of the most recent pioneers in that section of the field was able to apply fractal mathematics to the field of economics. One of his first discoveries was finding a fractal pattern to fluctuations in cotton prices. What are some other applications of fractals to economics? (Obviously they may be too many or too few -but an answer collecting references and pointers to some of them is worthwhile, nevertheless).
Mandelbrot's original application to cotton prices, The Variation of Certain Speculative Prices, claimed that the price fluctuations followed a Lévy stable distribution with a stability parameter $\alpha$ below 2. ($\alpha=2$ corresponds to the normal distribution.)
To be honest, I'm not sure how this was an application of "fractals" where previous contributions were not. While the sample paths of the Lévy stable process we was investigating have fractal characteristics—for instance, a non-integer fractal dimension—so does good old Brownian motion, which has fractal dimension of 1.5. Indeed, the relationship between fractal dimension $D$ and the stability parameter $\alpha$ is $D=2-1/\alpha$, so that Lévy stable processes with $\alpha\in (1,2)$ actually have a $D$ closer to 1 than Brownian motion—in a sense, they're less fractal!
The contribution of Mandelbrot here was, I think, to illustrate the rich variety of possible stochastic processes that asset prices can follow. (Processes that happen to exhibit a variety of fractal features.) He proposed, for instance, both Lévy stable processes as in the cotton price paper and, later, fractional Brownian motion.
Since this isn't my field (and I'm relying on half-remembered undergraduate knowledge at the moment), I can't say too much about the cutting edge of research here. I do know that "tempered" Lévy stable distributions are still quite popular for modeling. (These distributions make tails enough thinner that all moments exist, whereas the usual Lévy stable distributions with parameter $\alpha$ have no moments $\alpha$ or higher. Not having a finite second moment is a big problem for financial applications, which is why this modification is necessary.) Variants of fractional Brownian motion remain common as well. This Wikipedia article, although not too well-written, discusses why such stochastic processes can be useful for financial modeling.
It's safe to say that Mandelbrot has a very, very active legacy in finance.
I did not read the following paper, but it seems to be using fractals in a game theoretic context :