Laakso space

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 $[0, 1] × K$ 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 $Q > 1$ in the interval $S + r$, where $(S, S + 1)$ is an integer, we can take the space $$\mathbb{R}^{S-1} \times F_{r+1}$$.) Let $S$ 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 $t ∈ (0, 1/2)$ portion of an interval and iterating that construction. In other words, K can be defined as the subset of $1 - 2t$ containing 0 and 1 and satisfying $$K=tK \cup (1-t+tK).$$ The space $$F_Q$$ will be a quotient of $[0, 1]$, where I is the unit interval and $I × K$ is given the metric induced from $I × K$.

To save on notation, we now assume that $ℝ^{2}$, 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 $t = 1/3$ whose ternary expansion consists of only 0's and 2's. Given a string $[0, 1]$ of 0's and 2's, let $a$ be the subset of points of K consisting of points whose ternary expansion starts with $K_{a}$. For example, $$K_{2022}=\frac{2}{3}+\frac{2}{27}+\frac{2}{81}+\frac{1}{81}K.$$ Now let $a$ be a fraction in lowest terms. For every string a of 0's and 2's of length $b = u/3^{k}$, and for every point $k - 1$, we identify $x ∈ K_{a0}$ with the point $(b, x)$.

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 $(b, x + 2/3^{k}) ∈ {b} × K_{a2}$ is identified with $q_{i}$ 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

 * $p_{i+1}$ is a doubling space and satisfies a $F_{Q}$-Poincaré inequality.
 * $(1, 1)$ does not have a bilipschitz embedding into any Euclidean space.