Laakso space

Template:Short description In mathematical analysis and metric geometry, Laakso spaces are a class of metric spaces which are fractal, in the sense that they have non-integer Hausdorff dimension, but that admit a notion of differential calculus. They are constructed as quotient spaces of Template:Math where K is a Cantor set.

Background

Cheeger defined a notion of differentiability for real-valued functions on metric measure spaces which are doubling and satisfy a Poincaré inequality, generalizing the usual notion on Euclidean space and Riemannian manifolds. Spaces that satisfy these conditions include Carnot groups and other sub-Riemannian manifolds, but not classic fractals such as the Koch snowflake or the Sierpiński gasket. The question therefore arose whether spaces of fractional Hausdorff dimension can satisfy a Poincaré inequality. Bourdon and Pajot were the first to construct such spaces. Tomi J. Laakso gave a different construction which gave spaces with Hausdorff dimension any real number greater than 1. These examples are now known as Laakso spaces.

Construction

We describe a space $F_Q$ with Hausdorff dimension $Q\in (1,2)$. (For integer dimensions, Euclidean spaces satisfy the desired condition, and for any Hausdorff dimension Template:Math in the interval Template:Math, where Template:Math is an integer, we can take the space ${ℝ}^{S-1}\times F_{r+1}$.) Let Template:Math be such that $$Q=1+\frac{\ln 2}{\ln (1/t)}.$$ Then define K to be the Cantor set obtained by cutting out the middle Template:Math portion of an interval and iterating that construction. In other words, K can be defined as the subset of Template:Math containing 0 and 1 and satisfying $$K=tK\cup (1-t+tK).$$ The space $F_Q$ will be a quotient of Template:Math, where I is the unit interval and Template:Math is given the metric induced from Template:Math.

To save on notation, we now assume that Template:Math, so that K is the usual middle thirds Cantor set. The general construction is similar but more complicated. Recall that the middle thirds Cantor set consists of all points in Template:Math whose ternary expansion consists of only 0's and 2's. Given a string Template:Math of 0's and 2's, let Template:Math be the subset of points of K consisting of points whose ternary expansion starts with Template:Math. For example, $$K_{2022}=\frac{2}{3}+\frac{2}{27}+\frac{2}{81}+\frac{1}{81}K.$$ Now let Template:Math be a fraction in lowest terms. For every string a of 0's and 2's of length Template:Math, and for every point Template:Math, we identify Template:Math with the point Template:Math.

We give the resulting quotient space the quotient metric: $$d_{F_Q}(p,q)=\inf (d_{I\times K}(p,q_1)+d_{I\times K}(p_2,q_2)+\cdots +d_{I\times K}(p_{n-1},q_{n-1})+d_{I\times K}(p_n,q)),$$ where each Template:Math is identified with Template:Math and the infimum is taken over all finite sequences of this form.

In the general case, the numbers b (called wormhole levels) and their orders k are defined in a more complicated way so as to obtain a space with the right Hausdorff dimension, but the basic idea is the same.

Properties

• Template:Math is a doubling space and satisfies a Template:Math-Poincaré inequality.
• Template:Math does not have a bilipschitz embedding into any Euclidean space.