Realization of Metric Spaces as Inverse Limits, and Bilipschitz Embedding in L₁

We give sufficient conditions for a metric space to bilipschitz embed in L₁. In particular, if X is a length space and there is a Lipschitz map u: X → ℝ such that for every interval I ⊂ ℝ, the connected components of u^-1(I) have diameter ≤ const} ̇ diam(I), then X admits a bilipschitz embedding in L₁. As a corollary, the Laakso examples, (Geom Funct Anal 10(1):111-123, 2000), bilipschitz embed in L₁, though they do not embed in any any Banach space with the Radon-Nikodym property (e. g. the space ℓ₁ of summable sequences). The spaces appearing the statement of the bilipschitz embedding theorem have an alternate characterization as inverse limits of systems of metric graphs satisfying certain additional conditions. This representation, which may be of independent interest, is the initial part of the proof of the bilipschitz embedding theorem. The rest of the proof uses the combinatorial structure of the inverse system of graphs and a diffusion construction, to produce the embedding in L₁.