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Laakso space

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Type of mathematical fractal space

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 × 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 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 ℝ.

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 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/3 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 + 2/3) ∈ {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

References

  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.
  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.
  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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