Convolution Equations on Lattices: Periodic Solutions with Values in a Prime Characteristic Field

Abstract

These notes are inspired by the theory of cellular automata. The latter aims, in particular, to provide a model for inter-cellular or inter-molecular interactions. A linear cellular automaton on a lattice Λ is a discrete dynamical system generated by a convolution operator Δ _a : f → f * a with kernel a concentrated in the nearest neighborhood ω of 0 in Λ. In [Za1] we gave a survey (limited essentially to the characteristic 2 case) on the σ ⁺-cellular automaton with kernel the constant function 1 in ω. In the present paper we deal with general convolution operators over a field of characteristic p > 0. Our approach is based on the harmonic analysis. We address the problem of determining the spectrum of a convolution operator in the spaces of pluri-periodic functions on Λ. This is equivalent to the problem of counting points on the associate algebraic hypersurface in an algebraic torus according to their torsion multi-orders. These problems lead to a version of the Chebyshev-Dickson polynomials parameterized this time by the set of all finite index sublattices of Λ and not by the naturals as in the classical case. It happens that the divisibility property of the classical Chebyshev-Dickson polynomials holds in this more general setting.

Acknowledgements. This work partially was done during the author’s visit to the MPIM at Bonn. The author thanks this institution for a generous support and excellent working conditions.