Spatially chaotic configurations and functional equations with rescaling

Gregory Derfel, Rolf Schilling

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


The functional equation y(qt) = 1/4q[y(t + 1) + y(t - 1) + 2y(t)] (0 < q < 1) t ∈ ℝ is associated with the appearance of spatially chaotic structures in amorphous (glassy) materials. Continuous compactly supported solutions of the above equation are of special interest. We shall show that there are no such solutions for 0 < q < 1/2, whereas such a solution exists for almost all 1/2 < q < 1. The words 'for almost all q' in the previous sentence cannot be omitted. There are exceptional values of q in the interval [1/2, 1] for which there are no integrable solutions. For example, q = (√5 - 1)/2 ≈ 0.618, which is the reciprocal of the 'golden ratio' is such an exceptional value. More generally, if λ is any Pisot-Vijayaraghavan number, or any Salem number, then q = λ-1 is an exceptional value.

Original languageEnglish
Pages (from-to)4537-4547
Number of pages11
JournalJournal of Physics A: Mathematical and General
Issue number15
StatePublished - 1 Dec 1996

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy


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