In mathematical analysis and metric geometry, Laakso spaces^[1][2] 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.^[3]

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^[4] were the first to construct such spaces. Tomi J. Laakso^[3] 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 ${\displaystyle F_{Q}}$ with Hausdorff dimension ${\displaystyle Q\in (1,2)}$. (For integer dimensions, Euclidean spaces satisfy the desired condition, and for any Hausdorff dimension S + r in the interval (S, S + 1), where S is an integer, we can take the space ${\displaystyle \mathbb {R} ^{S-1}\times F_{r+1}}$.) Let t ∈ (0, 1/2) be such that

$${\displaystyle Q=1+{\frac {\ln 2}{\ln(1/t)}}.}$$

Then define K to be the Cantor set obtained by cutting out the middle 1 - 2t portion of an interval and iterating that construction. In other words, K can be defined as the subset of [0, 1] containing 0 and 1 and satisfying

$${\displaystyle K=tK\cup (1-t+tK).}$$

The space ${\displaystyle F_{Q}}$ will be a quotient of I × K, where I is the unit interval and I × K is given the metric induced from ℝ².

To save on notation, we now assume that t = 1/3, 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 [0, 1] whose ternary expansion consists of only 0's and 2's. Given a string a of 0's and 2's, let K_a be the subset of points of K consisting of points whose ternary expansion starts with a. For example,

$${\displaystyle K_{2022}={\frac {2}{3}}+{\frac {2}{27}}+{\frac {2}{81}}+{\frac {1}{81}}K.}$$

Now let b = u/3^k be a fraction in lowest terms. For every string a of 0's and 2's of length k - 1, and for every point x ∈ K_a0, we identify (b, x) with the point (b, x + 2/3^k) ∈ {b} × K_a2.

We give the resulting quotient space the quotient metric:

$${\displaystyle d_{F_{Q}}(p,q)=\inf(d_{I\times K}(p,q_{1})+d_{I\times K}(p_{2},q_{2})+\cdots +d_{I\times K}(p_{n-1},q_{n-1})+d_{I\times K}(p_{n},q)),}$$

where each q_i is identified with p_i+1 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

• F_Q is a doubling space and satisfies a (1, 1)-Poincaré inequality.
• F_Q does not have a bilipschitz embedding into any Euclidean space.

References