Growth behaviour of periodic tame friezes

Abstract

We examine the growth behaviour of the entries occurring in $n$-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.