Gunther Cornelissen: Dynamics in algebraic groups and cellular automata

Abstract:

Some elementary counting problems, such as determining the cardinality of the group of invertible matrices of given size over a given finite field, can be recast in terms of dynamics of endomorphisms of algebraic groups (in this case, the action of Frobenius). For semisimple groups, Steinberg famously computed fixed points of general endomorphisms in his 1968 memoir. We generalize this to arbitrary algebraic groups, where we see the appearance of p-adic fluctuations in the fixed point formula, irrationality of the zeta function, non-hyperbolic behaviour in the orbit distribution, etc. We recognised one of the formulas from the theory of cellular automata, which led us to prove an actual correspondence of spaces that explains the numerical coincidence. A consequence is that all our new results on algebraic groups apply directly to describe the (hitherto unknown) dynamical properties of certain (“linear multiband”) cellular automata. (Joint work with Jakub Byszewski and Marc Houben).