smoketrail lacunaries

 

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Erik Nelson noted that the plumes of exponents kind of look like smoke trails, so I’m going to appropriate that as terminology. These are smoketrail lacunaries. What’s interesting here is that we’re not looking at an escape time or Newton’s method fractal per se, but there are features of these smoketrail lacunaries that are on the way to being Cantor Bouquets. For a good introduction to Cantor Bouquets, read Bob Devaney’s http://www.ams.org/notices/200401/fea-devaney.pdf Cantor and Sierpinski, Julia and Fatou: Complex Topology Meets Complex Dynamics, Notices of the AMS)

Long term complex analysis plans

Eventually, I’d like to be able to do real time complex analysis. Here’s a sketch of the plan: 

The interface is going to be a didgeridoo hooked up to a microphone. (there’s a nice synthesis of the rainbow in Australian mythology with the didgeridoo, so that’s a bonus). The microphone will be attached to a computer with an asic that performs real time special functions computation (even a GPU is going to feature special functions in software, and that’s going to get really bogged down if I’m defining new things on the fly, so hardware it must be). And the machine will generate phase portraits in real time, showing the LaTeX’d expression for the function whose phase portrait is on the screen at the moment. 

The idea here is that lacunaries (or any function on the complex plane) aren’t like butterflies or stamps: although you could make a picture book showing lacunaries with their series or product representations, that would be a far cry from having a visual/kinesthetic/haptic grasp of how the visual appearance of the function depended on its definition. This is the kind of mathematics I’d enjoy doing, not fiddling around with exact sequences here or there, or whinging on about cocomplete subfunctors on pseudorigid categories. I’m not here to play woefully unsatisfying language games.