With this paper, the authors continue their series of papers on metric measure spaces which satisfy a doubling condition and a Poincaré inequality; the preceding paper in the series is [J. Cheeger and B. Kleiner, Geom. Funct. Anal. 23, No. 1, 96–133 (2013; Zbl 1277.46012)].
The authors consider inverse systems of connected metric measure graphs, \[ X_0 \overset{\pi_0}\longleftarrow \cdots\overset{\pi_{i-1}}\longleftarrow X_{i} \overset{\pi_i}\longleftarrow\cdots \,. \] The main result of the paper asserts that if the inverse system satisfies the conditions whose short description is presented below, its Gromov-Hausdorff limit is a doubling metric space satisfying a Poincaré inequality. The authors also prove that (except for some degenerate cases) such spaces do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property. Results of [loc.cit.]imply that such spaces admit bilipschitz embeddings into \(L_1\).
The authors consider systems of metric connected graphs \(\{X_i\}\) endowed with path metrics \(d_i\) and measures \(\mu_i\) such that for some constants \(2\leq m\in\mathbb{Z}\), \(\Delta, \theta\in (0,\infty)\) and every \(i\in \mathbb{Z}\), the following conditions hold:

(1)

\((X_i,d_i)\) is such that the degrees of all vertices are \(\leq \Delta\), and every edge of \( X_i\) is isometric to an interval of length \(m^{-i}\) with respect to the path metric \(d_i\).

(2)

If \(X_i'\) denotes the graph obtained by subdividing each edge of \(X_i\) into \(m\) edges of length \(m^{-(i+1)}\), then \(\pi_i\) induces a map \(\pi_i:(X_{i+1},d_{i+1})\to (X_i',d_i)\) which is open, simplicial, and an isometry on every edge.

(3)

For every \(x_i\in X_i'\), the inverse image \(\pi_i^{-1}(x_i)\subset X_{i+1}\) has \(d_{i+1}\)-diameter at most \(\theta \cdot m^{-(i+1)}\).

(4)

The measure \(\mu_i\) restricts to a constant multiple of arc length on each edge \(e_i\subset X_i\), and satisfies certain additional conditions.