6533b7d8fe1ef96bd126a5bc

RESEARCH PRODUCT

Assouad dimension, Nagata dimension, and uniformly close metric tangents

Tapio RajalaEnrico Le Donne

subject

Pure mathematicssub-Riemannian manifoldsGeneral Mathematics54F45 (Primary) 53C23 54E35 53C17 (Secondary)01 natural sciencessymbols.namesakeMathematics - Geometric TopologyDimension (vector space)Mathematics - Metric Geometry0103 physical sciencesFOS: MathematicsMathematics (all)assouad dimensionMathematics::Metric GeometryPoint (geometry)0101 mathematicsMathematics010102 general mathematicsta111TangentMetric Geometry (math.MG)Geometric Topology (math.GT)16. Peace & justiceMetric dimensionAssouad dimension; Metric tangents; Nagata dimension; Sub-Riemannian manifolds; Mathematics (all)Metric spaceBounded functionNagata dimensionMetric (mathematics)symbols010307 mathematical physicsMathematics::Differential Geometrymetric tangentsLebesgue covering dimension

description

We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed in addition is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our results to equiregular subRiemannian manifolds and show that locally their Nagata dimension equals the topological dimension.

yearjournalcountryeditionlanguage
2013-06-25
http://arxiv.org/abs/1306.5859