Growth behaviour of periodic tame friezes

TL;DR

This paper investigates the growth patterns of entries in periodic tame friezes, establishing recursive relations and analyzing their asymptotic behavior, especially for friezes derived from geometric triangulations.

Contribution

It extends previous work by proving recursive relations for all entries, introduces growth coefficients, and characterizes their behavior for specific geometric configurations.

Findings

Entries grow asymptotically exponentially in certain friezes.

Growth coefficients are determined by a principle coefficient read from the frieze.

Friezes from punctured discs exhibit linear growth.

Abstract

We examine the growth behaviour of the entries occurring in $n$-periodic tame friezes of real numbers. Extending \cite{T}, 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 are shown to satisfy itself a recursive relation. Thus, all growth coefficients are determined by a \emph{principle growth coefficients}, 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.