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]
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.
We describe a space with Hausdorff dimension . (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 .) Let t ∈ (0, 1/2) be such that
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,
We give the resulting quotient space the quotient metric:
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.