Growth behaviour of periodic tame friezes
Growth behaviour of periodic tame friezes
复制标题
周期性驯服饰带的生长行为
DOI：
10.4171/rmi/1063
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Manuela Tschabold
中科院分区：
文献类型：
--
作者：
K. Baur;K. Fellner;M. J. Parsons;Manuela Tschabold
我们考察了出现在实数的$n$周期驯服浮雕中的项的增长行为。在引证{T}的基础上，证明了这类图的所有表项之间存在广义递归关系。这些递归由一系列所谓的增长系数来参数化，这些系数被证明满足其自身的递归关系。因此，所有的生长系数都是由主生长系数决定的，它可以直接从板条上读出。
我们特别强调了正整数的周期性驯服褶皱，指定了增长系数对于任何这样的褶皱所取的值。我们证明了由环的三角剖分产生的这对褶皱的增长系数是一致的。两者的条目都被显示为渐近指数增长，而被刺穿的圆盘的三角剖分仅提供了线性增长的褶皱。
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.