Analysis on Graphs and Fractals

Pluri-periods of harmonic functions on Caley graphs

M Zaidenberg, Fourier Institute, Grenoble, France

Abstract

We consider a Caley graph $\Gamma$ of a free abelian group (i.e. a lattice), and harmonic functions on $\Gamma$ with values in a field $K$ of positive characteristic. We are interested in pluri-periodic such functions. Which pluri-periods can occur? This question arises naturally (in characteristic 2 case) in relation with the game "Lights out" on a square greed, or otherwise in the dynamics of linear cellular automata on lattices. We will present two kind of answers. The first one is related to Chebyshev-Dickson polynomials and their generalizations. The second one is related to points count on certain affine algebraic varieties over $K$, or, equivalently, to determining harmonic characters on lattices.