In the 1970's Ernst Kunz attached a function to each characteristic p local ring - the function is similar to the Hilbert-Samuel function, but uses "Frobenius powers" of the maximal ideal rather than ordinary powers. I've been exploring this function (which I call the Hilbert-Kunz function) for some 10 years.

Fractals and characteristic p dynamical systems appear to turn up in these investigations. Teixeira and I now have a theory of "fractal patchwork quilts" that allows us to compute various Hilbert-Kunz functions, that allows us to explicitly write down the dynamical systems that arise, and that in its elegance and strangeness has an independent interest. My present research interests center mostly on these topics.