The Analyst's traveling salesman theorem in graph inverse limits
The Analyst's traveling salesman theorem in graph inverse limits
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图逆极限中的分析师旅行商定理
DOI：
10.5186/aasfm.2017.4260
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Raanan Schul
中科院分区：
文献类型：
--
作者：
Guy C. David;Raanan Schul
在一类由Laakso引入并由Cheeger-Kleiner推广的高度非欧度量空间中，证明了Peter Jones分析师的旅行商定理的一个版本。这些空间被构造为度量图的逆极限，并且包括加倍且具有Poincare不等式的例子。我们证明了其中一个空间中的集合包含在一条可校正的曲线中当且仅当它在大多数位置和尺度上是定量“平坦”的，其中平坦度是相对于所谓的单调测地线来测量的。这提供了对这些空间内的定量可纠正性的第一次检验。
We prove a version of Peter Jones' Analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger-Kleiner. These spaces are constructed as inverse limits of metric graphs, and include examples which are doubling and have a Poincare inequality. We show that a set in one of these spaces is contained in a rectifiable curve if and only if it is quantitatively "flat" at most locations and scales, where flatness is measured with respect to so-called monotone geodesics. This provides a first examination of quantitative rectifiability within these spaces.