Path-homotopy is equivalent to $\mathbb{R}$-tree reduction

Abstract

Suppose a path $α$ factors through an $\mathbb{R}$-tree $T$ as $α=q \circ p $. Let $r$ parameterize the unique geodesic in $T$ joining the endpoints of $p$. Then we say that the path $β=q \circ r$ is obtained from $α$ by "geodesic $\mathbb{R}$-tree reduction." Essentially, $β$ is obtained from $α$ by deleting one-dimensional back-tracking. In this paper, we show that any two homotopic paths are geodesic $\mathbb{R}$-tree reductions of some single common path. Hence, the equivalence relation on paths generated by geodesic $\mathbb{R}$-tree reduction is precisely path-homotopy. The common path is explicitly constructed and is necessarily space-filling in the image of a given path-homotopy.

Figures (6)

• Figure 1: An illustration of part of the homeomorphism $f:[0,1]\to[0,1]$ which maps $I_n\to K_n$ and $J_n\to L_n$ in a piecewise-linear fashion.
• Figure 2: A CIP-loop is an inverse pair loop which either has the form of an inverse pair $\alpha\overline{\alpha}$ of Cantor paths (above) or of the form $\alpha_1 \beta \overline{\alpha_1}$ for a Cantor path $\alpha_1$ and constant path $\beta$ (below).
• Figure 3: Since $L_{x_1,y_1}$ and $\overline{L_{y_2,x_2}}$ are both parameterized by $\tau$, their metric distance is the maximum length between starting and ending points.
• Figure 4: The upper segment illustrates the selected sets $A_n$ and $B_{n,j}$ on which $\alpha$ is constant. The lower segment illustrates the selected sets $C_n$ and $D_{n,j}$ on which $\beta$ is constant.
• Figure 5: The interlocking pattern that determines the structure of $\alpha '$ and $\beta '$. The triangles represent inserted inverse-pair loops and the trapezoids represent an inverse pair loop but with a constant path included in the middle.

Theorems & Definitions (46)

• Corollary 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Definition 3.1
• Definition 3.2
• Definition 3.3