Geometry and Dynamics of Groups and Spaces pp 721-742 | Cite as

# 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.

## Keywords

Cellular automaton Chebyshev-Dickson polynomial convolution operator lattice sublattice finite field discrete Fourier transform discrete harmonic function pluri-periodic function## Preview

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