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 $[0, 1] \times K$ 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 $F_{Q}$ with Hausdorff dimension $Q\in (1,2)$ . (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 $\mathbb {R} ^{S-1}\times F_{r+1}$ .) Let $t \in (0, 1/2)$ be such that $Q=1+{\frac {\ln 2}{\ln(1/t)}}.$ Then define K to be the Cantor set obtained by cutting out the middle $1 - 2 t$ 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 $K=tK\cup (1-t+tK).$ The space $F_{Q}$ will be a quotient of $I \times K$ , where I is the unit interval and $I \times K$ is given the metric induced from $ℝ 2$ .

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, $K_{2022}={\frac {2}{3}}+{\frac {2}{27}}+{\frac {2}{81}}+{\frac {1}{81}}K.$ 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 $x \in K a 0$ , we identify $(b, x)$ with the point $(b, x + 2/3 k) \in {b} \times K a 2$ .

We give the resulting quotient space the quotient metric: $d_{F_{Q}}(p,q)=\inf(d_{I\times K}(p,q_{1})+d_{I\times K}(p_{2},q_{2})+\cdots +d_{I\times K}(p_{n-1},q_{n-1})+d_{I\times K}(p_{n},q)),$ 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

^ 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.
^ ^a ^b 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. arXiv:math/9710208. doi:10.1090/S0002-9939-99-04901-1.

[1] 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.

[Hei-2] 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.

[Laakso-3] 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.

[4] 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. arXiv:math/9710208. doi:10.1090/S0002-9939-99-04901-1.

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