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

  • Mikhail Zaidenberg
Chapter
Part of the Progress in Mathematics book series (PM, volume 265)

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 : ff * 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 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Mikhail Zaidenberg
    • 1
  1. 1.Institut Fourier, UMR 5582 CNRS-UJFUniversité Grenoble ISt. Martin d’Hères cédexFrance

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