2006/12/18 by Marius Buliga, Buliga, Marius · 1 citation
Mathematics · #20F65 #22A10 #51F99 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #math.GR #math.MG #msc:20F65 #msc:22A10 #msc:51F99
paper · pdf · doi:10.48550/arxiv.math/0612509
arxiv created 2007/02/14 · arxiv updated 2009/12/01
In this paper we continue the study of dilatation structures, introduced in math.MG/0608536 . A dilatation structure on a metric space is a kind of enhanced self-similarity. By way of examples this is explained here with the help of the middle-thirds Cantor set. Linear and self-similar dilatation structures are introduced and studied on ultrametric spaces, especially on the boundary of the dyadic tree (same as the middle-thirds Cantor set). Some other examples of dilatation structures, which share some common features, are given. Another class of examples, coming from sub-Riemannian geometry, will make the subject of an article in preparation. In the particular case of ultrametric spaces the axioms of dilatation structures take a simplified form, leading to a description of all possible weak dilatation structures on the Cantor set. As an application we prove that there is more than one linear and self-similar dilatation structure on the Cantor set, compatible with the iterated functions system which defines the Cantor set. Applications to self-similar groups are reserved for a further paper.