Laakso space
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 [math]\displaystyle{ F_Q }[/math] with Hausdorff dimension [math]\displaystyle{ Q \in (1,2) }[/math]. (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 [math]\displaystyle{ \mathbb{R}^{S-1} \times F_{r+1} }[/math].) Let t ∈ (0, 1/2) be such that [math]\displaystyle{ Q=1+\frac{\ln 2}{\ln(1/t)}. }[/math] 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 [math]\displaystyle{ K=tK \cup (1-t+tK). }[/math] The space [math]\displaystyle{ F_Q }[/math] 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, [math]\displaystyle{ K_{2022}=\frac{2}{3}+\frac{2}{27}+\frac{2}{81}+\frac{1}{81}K. }[/math] 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: [math]\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)), }[/math] 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
- ↑ Heinonen, Juha; Koskela, Pekka; Shanmugalingam, Nageswari; Tyson, Jeremy T. (2015). Sobolev spaces on metric measure spaces: an approach based on upper gradients. Cambridge University Press. p. 403. ISBN 9781107092341.
- ↑ Heinonen, Juha (24 January 2007). "Nonsmooth calculus". Bulletin of the American Mathematical Society 44 (2): 163–232. doi:10.1090/S0273-0979-07-01140-8.
- ↑ 3.0 3.1 Laakso, T.J. (1 April 2000). "Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality". Geometric and Functional Analysis 10 (1): 111–123. doi:10.1007/s000390050003.
- ↑ Bourdon, Marc; Pajot, Hervé (9 April 1999). "Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings". Proceedings of the American Mathematical Society 127 (8): 2315–2324. doi:10.1090/S0002-9939-99-04901-1.
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