Spectral Domain Boundaries in Cellular Automata

  • Authors:

  • Marcus Pivato

  • Affiliations:

  • Dept. of Mathematics & Computer Science, Wesleyan University, Middletown CT, USA and Dept. of Mathematics, Trent University, Peterborough, Ontario, Canada. E-mail: pivato@xaravve.trentu.ca

  • Venue:
  • Fundamenta Informaticae - Special issue on DLT'04
  • Year:
  • 2007

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Abstract

Let A^{Z^D} be the Cantor space of Z^D-indexed configurations ina finite alphabet A, and let σ be the Z^D-action of shifts onA^{Z^D}. A cellular automaton is a continuous, σ-commutingself-map Φ of A^{Z^D}, and a Φ-invariant subshift is aclosed, (Φ, σ)-invariant subset u ⊂ A^{Z^D}. Supposea ε A^{Z^D} is u-admissible everywhere except for somesmall region we call a defect. It has been empirically observedthat such defects persist under iteration of Φ, and oftenpropagate like 'particles' which coalesce or annihilate on contact.We use spectral theory to explain the persistence of some defectsunder Φ, and partly explain the outcomes of theircollisions.