Quasisymmetric rectifiability of uniformly disconnected sets

Abstract

We prove that uniformly disconnected subsets of metric measure spaces with controlled geometry (complete, Ahlfors regular, supporting a Poincare inequality, and a mild topological condition) are contained in a quasisymmetric arc. This generalizes a result of MacManus in 1999 from Euclidean spaces to abstract metric setting. Along the way, we prove a geometric strengthening of the classical Denjoy-Riesz theorem in metric measure spaces. Finally, we prove that the complement of a uniformly disconnected set in such a metric space is uniform, quantitatively.

Figures (5)

• Figure 1: In this example, $w$ has an even number of letters, $n_w = 2$ and $n_{w1}=n_{w2}=2$. The blue curves are the image of $h_{w}$ defined in §\ref{['sec:even']} while the red curves are part of the image of $h_{w^{\uparrow}}$ defined in §\ref{['sec:odd']}.
• Figure 2: The positions of $\hat{a}_{ui',j'}$, $\hat{b}_{ui',j'}$, $\hat{a}_{wi,j}$, and $\hat{b}_{wi,j}$. Note that $b_{ui',j'} = a_{wi,j}$.
• Figure 3: The positions of $\widetilde{b}_{ui',j'}$, $t_{ui',j'}^R$, $t_{wi,j}^L$, and $\widetilde{a}_{wi,j}$.
• Figure 4: In this figure, the red curves are the images of $h_u|\hat{I}_{ui',j'}$ (left) and $h_w|\hat{I}_{wi,j}$ (right), the blue curve is the curve $\Gamma_{(ui',j'),(wi,j)}$ and the green curves are the geodesics $g^R_{ui',j'}$ (left) and $g^L_{wi,j}$ (right).
• Figure 5: The positions of $\hat{a}_{wi,j}$, $\widetilde{a}_{wi,j}$, $\widetilde{b}_{wi,j}$, and $\hat{b}_{wi,j}$.

Theorems & Definitions (48)

• Theorem 1.1: M99
• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Lemma 2.1: H01
• Lemma 2.2: BHR01
• Lemma 2.3: H01
• Lemma 2.4: H01
• Lemma 2.5: HK98
• Lemma 2.6: C99