Laakso space explained

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 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^[3] were the first to construct such spaces. Tomi J. Laakso^[4] 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

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

To save on notation, we now assume that, 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 whose ternary expansion consists of only 0's and 2's. Given a string of 0's and 2's, let be the subset of points of K consisting of points whose ternary expansion starts with . For example,$$K\_=\backslash frac+\backslash frac+\backslash frac+\backslash fracK.$$Now let be a fraction in lowest terms. For every string a of 0's and 2's of length, and for every point, we identify with the point .

We give the resulting quotient space the quotient metric:$$d\_(p,q)\; =\; \backslash inf(d\_(p,q\_1)+d\_(p\_2,q\_2)+\backslash cdots+d\_(p\_,q\_)+d\_(p\_n,q)),$$where each is identified with 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

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

Notes and References

1. Book: Heinonen . Juha . Juha Heinonen . Koskela . Pekka . Shanmugalingam . Nageswari . Tyson . Jeremy T. . Sobolev spaces on metric measure spaces: an approach based on upper gradients . 2015 . Cambridge University Press . 9781107092341 . 403 . HKST.
2. Heinonen . Juha . Juha Heinonen . Nonsmooth calculus . . 24 January 2007 . 44 . 2 . 163–232 . 10.1090/S0273-0979-07-01140-8 . free.
3. Bourdon . Marc . Pajot . Hervé . Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings . . 9 April 1999 . 127 . 8 . 2315–2324 . 10.1090/S0002-9939-99-04901-1 . free. math/9710208 .
4. Laakso . T.J. . Ahlfors Q-regular spaces with arbitrary admitting weak Poincaré inequality . . 1 April 2000 . 10 . 1 . 111–123 . 10.1007/s000390050003. free.