I see you asking for asks youre about to get it
Your blog has fractals in the name but I haven't seen you talk about fractals the past year, can you tell us something about the math behind fractals that you really like?
Of course!!
The math I'm currently doing is mostly ergodic theory, which is what you get from doing measurable dynamics (i.e. repeatable transformations that map a set to itself such that any inverse image of a measurable set is measurable) and thinking about the "statistical dynamic properties". Statistical in this sense has little to do with actual statistics. Instead, it means we look at the properties of almost every point in the set, and don't worry about the rest.
An example is the doubling map, which takes a real number x between 0 and 1 (inclusive) and maps it to 2x mod 1 (so you remove the integer part, meaning both 0.25 and 0.75 are mapped to 0.5). If you keep applying this map on (almost) any x (0.2 -> 0.4 -> 0.8 -> 0.6 etc) and keep track of where it goes, it will spend about half of the time inbetween 0.5 and 1. Of course, 0 is mapped to 0 and so it never gets there under the doubling map, but with ergodic theory we can just sweep it under the rug.
Something I really like about fractals is that they can be defined this way! Whenever someone doing ergodic/measure theory says something is true "almost everywhere", fractal theorists will ask the question "okay but what about the exceptions?" and get awesome shapes out of that. So in a way, these fields complement each other perfectly!
