The first, maybe the simplest is called "Fractals", and when you use it, it draws little arrows, some pointing up, others pointing down, like this:
This indicator, however, has nothing to do with fractals, it relates to Elliot Wave Theory, as explained here: http://trading-stocks.netfirms.com/fractals.htm
A derivation of this is called the "fractal channel" which links the little arrows, similarly, it has nothing to do with fractals.
More relevant then is the Fractal Adaptive Moving Average, which relates to Kaufman's AMA, but uses fractal theory to determine the current volatility of the market in order to adjust the speed of the MA. The idea of the AMA is to slow down the MA when the market is moving sideways, and to speed it up when there is a trend. To achieve this objective, John Ehlers developped the FRAMA, using the Fractal Dimension as a direct measurement of Volatility, he explains his method in a file (title: FRAMA) that can be downloaded from this address: http://www.mesasoftware.com/technicalpapers.htm
On the following graph, I plotted a simple 16-MA (blue), an exponential 16-MA (yellow) and the FRAMA in red (with a reference period of 16 as well). Below are the fractal dimension used by the FRAMA (and computed from the formula of the above paper), as well as a more sophisticated fractal dimension (to which I will come later):
Clearly, during the sideways market (until about 16:45), the FRAMA is somewhat smoothier than the two others, and when the trend goes on, it also reacts faster. Therefore, we can say that the FRAMA is a good AMA. However, it could be better, the computation of the fractal dimension is rough to say the least, it oscillates between extreme values (from 2 to below 1) that don't even make sense mathematically. The FDI plotted in the lowest window, displays a more reasonable fractal dimension (the period to calculate both is 16), for those interested in this tool, I would therefore advise to use the FDI and that might entail a modification in the factor -4.6 in the computation of the coefficient alpha (from the FRAMA paper) where Ehlers recommends:
The fractal dimension Df in the FDI follows the following formula:
Where N is the number of periods(price valuations) considered. Df provides us with some idea of volatility, when Df gets close to 2, it means that we have very high volatility, the closer to 1 and we have low volatility or, in other terms, a well-defined trend. But that's very general qualitative comments, the passage to a computable quantity is trickier. Elhers assumes that price movements are following a lognormal distribution (which is not the case) and, on this basis, comes to compute the value of alpha as an exponential. I will, in the near future, share my reflections on how to get an identified numerical measure of entropy (volatility) from Df.
But for now, my point is merely to say that the fractal dimension is an indicator of volatility, it does not inform on the direction of the market. To get this direction, many analysts rely on MA or combination of them (such as Ichimoku, Bands,...), those indicators may be refined, using the fractal theory, but they then become hybrid indicators, mixing two diverging conceptions of what price movement is about.
As of now, and as far as I know, the only technical tool fractal theory is providing is a measure of volatility, but volatility in itself may be an interesting information to set up one's stop and position size. It may not be necessary to use volatility as a mere entry variable into another indicator.