JournalsrmiVol. 35, No. 2pp. 575–606

Growth behaviour of periodic tame friezes

  • Karin Baur

    Universität Graz, Austria and University of Leeds, UK
  • Klemens Fellner

    Universität Graz, Austria
  • Mark J. Parsons

  • Manuela Tschabold

Growth behaviour of periodic tame friezes cover
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Abstract

We examine the growth behaviour of the entries occurring in nn-periodic tame friezes of real numbers. Extending work of the last author, we prove that generalised recursive relations exist between all entries of such friezes. These recursions are parametrised by a sequence of so-called growth coefficients, which is itself shown to satisfy a recursive relation. Thus, all growth coefficients are determined by a principal growth coefficient, which can be read-off directly from the frieze.

We place special emphasis on periodic tame friezes of positive integers, specifying the values the growth coefficients take for any such frieze. We establish that the growth coefficients of the pair of friezes arising from a triangulation of an annulus coincide. The entries of both are shown to grow asymptotically exponentially, while triangulations of a punctured disc are seen to provide the only friezes of linear growth.

Cite this article

Karin Baur, Klemens Fellner, Mark J. Parsons, Manuela Tschabold, Growth behaviour of periodic tame friezes. Rev. Mat. Iberoam. 35 (2019), no. 2, pp. 575–606

DOI 10.4171/RMI/1063