[Paper Review] On the Assouad dimension and convergence of metric spaces

Yoshito Ishiki|arXiv (Cornell University)|Nov 18, 2019

Geometric Analysis and Curvature Flows11 references4 citations

TL;DR

This paper introduces pseudo-cones as a generalization of tangent and asymptotic cones in metric spaces, proving that the Assouad dimension of any metric space is bounded below by that of its pseudo-cones. It establishes that the Assouad dimension is stable under Gromov–Hausdorff limits and ultralimits, and constructs an $(\omega_0+1)$-metric space containing all compact metric spaces as pseudo-cones, demonstrating that analogies of such dimension bounds fail for topological, Hausdorff, and conformal Hausdorff dimensions.

ABSTRACT

We introduce the notion of pseudo-cones of metric spaces as a generalization of both of the tangent cones and the asymptotic cones. We prove that the Assouad dimension of a metric space is bounded from below by that of any pseudo-cone of it. We exhibit a example containing all compact metric spaces as pseudo-cones, and examples containing all proper length spaces as tangent cones or asymptotic cones.

Motivation & Objective

• To generalize the concept of tangent and asymptotic cones via pseudo-cones for studying metric space dimensions.
• To establish lower bounds for the Assouad dimension of a metric space using its pseudo-cones.
• To demonstrate that the Assouad dimension is preserved under Gromov–Hausdorff convergence and ultralimit constructions.
• To construct a single metric space whose pseudo-cones include all compact metric spaces, and another whose tangent or asymptotic cones include all proper length spaces.
• To show that analogies of Assouad dimension bounds do not hold for topological, Hausdorff, or conformal Hausdorff dimensions.

Proposed method

• Define pseudo-cones as Gromov–Hausdorff limits of scaled subsets $ u_i A_i $ of a metric space $ X $, where $ u_i \in (0,\infty) $.
• Use the stability of the Assouad dimension under metric scaling: $ \dim_A(hX) = \dim_A(X) $ for $ h > 0 $.
• Apply techniques from Mackay and Tyson (2010) on tangent spaces to extend lower bounds on Assouad dimension to pseudo-cones.
• Construct a metric space $ X $ using nested sets $ G_i $, with controlled distances and scaling factors $ a_i $, to ensure desired convergence properties.
• Utilize ultralimits via non-principal ultrafilters to analyze limit behavior of scaled subsets, proving dimension bounds for ultralimit pseudo-cones.
• Employ combinatorial and geometric estimates (e.g., $ \alpha(G_i)/\alpha(G_{i-1}) < 16 $) to control distances and convergence rates in the construction.

Experimental results

Research questions

• RQ1 Can the notion of tangent and asymptotic cones be generalized to a broader class of limit objects in metric spaces?
• RQ2 Does the Assouad dimension of a metric space bound the Assouad dimension of its pseudo-cones?
• RQ3 Can a single metric space be constructed such that all compact metric spaces appear as pseudo-cones?
• RQ4 Do dimension bounds based on tangent or asymptotic cones extend to conformal Assouad dimension?
• RQ5 Are analogies of Assouad dimension bounds valid for topological, Hausdorff, or conformal Hausdorff dimensions?

Key findings

• For any metric space $ X $, the Assouad dimension of every pseudo-cone $ P \in \mathrm{PC}(X) $ satisfies $ \dim_A P \leq \dim_A X $.
• The Assouad dimension of any ultralimit $ \lim_{\mathcal{U}}(u_i A_i, a_i) $ is bounded above by $ \dim_A X $.
• There exists an $ (\omega_0+1) $-metric space $ X $ such that $ \mathrm{PC}(X) $ contains all compact metric spaces as pseudo-cones.
• There exists an $ (\omega_0+1) $-metric space $ X $ for which every proper length space appears as a tangent or asymptotic cone.
• The conformal Assouad dimension satisfies $ \mathrm{Cdim}_A P \leq \mathrm{Cdim}_A X $ for every $ P \in \mathrm{KPC}(X) $, the class of compact pseudo-cones.
• The topological, Hausdorff, and conformal Hausdorff dimensions do not satisfy analogous lower bounds: there exist $ X $ and $ P \in \mathrm{PC}(X) $ such that $ \dim_T X < \dim_T P $, $ \dim_H X < \dim_H P $, and $ \mathrm{Cdim}_H X < \mathrm{Cdim}_H P $.