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 {\displaystyle F_{Q}} with Hausdorff dimension Q ∈ ( 1 , 2 ) {\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 R S − 1 × F r + 1 {\displaystyle \mathbb {R} ^{S-1}\times F_{r+1}} .) Let t ∈ (0, 1/2) be such that
Q = 1 + ln 2 ln ( 1 / t ) . {\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
K = t K ∪ ( 1 − t + t K ) . {\displaystyle K=tK\cup (1-t+tK).}
The space F Q {\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 Ka be the subset of points of K consisting of points whose ternary expansion starts with a. For example,
K 2022 = 2 3 + 2 27 + 2 81 + 1 81 K . {\displaystyle K_{2022}={\frac {2}{3}}+{\frac {2}{27}}+{\frac {2}{81}}+{\frac {1}{81}}K.}
Now let b = u/3k 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 ∈ Ka0, we identify (b, x) with the point (b, x + 2/3k) ∈ {b} × Ka2. We give the resulting quotient space the quotient metric:
d F Q ( p , q ) = inf ( d I × K ( p , q 1 ) + d I × K ( p 2 , q 2 ) + ⋯ + d I × K ( p n − 1 , q n − 1 ) + d I × K ( p n , q ) ) , {\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 qi is identified with pi+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 FQ is a doubling space and satisfies a (1, 1)-Poincaré inequality. FQ does not have a bilipschitz embedding into any Euclidean space.
References
