Uniform local radii, cyclic small-mesh polygons, mesh, length, energy, the midpoint operation and the zero-limit basin

Let $X$ be a compact locally CAT(1) metric space (Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles, Open cover, subcover, compact metric space, and compact subset of a metric space).

(1) Uniform local CAT(1) radius. A real number $l>0$ is a uniform local CAT(1) radius for $X$ if every closed ball $\bar{B}(x,r)\subseteq X$ with $x\in X$ and $0<r\le l$ (Open ball, closed ball and sphere in a metric space), with the induced metric, is a CAT(1) space. Such an $l$ exists and may be chosen with $l<\pi \mathrm{/}2$ (Pi is the first positive zero of sine); the existence proof is part of Existence of the uniform radius, continuity of the midpoint operation, the energy drop, its equality case, and convergence of zero-limit polygons ↗ and is not asserted by this definition. Fix such an $l$ for the remainder of the page.

(2) Cyclic tuples, mesh, length, energy. Fix an integer $n\ge 3$ and a real $h$ with $0<h<l$. For a tuple $x=(x_0,\dots ,x_{n-1})\in X^n$ the indices are read modulo $n$, and $\mathrm{mesh}(x):={\max }_{0\le i<n}d(x_i,x_{i+1}),L(x):={\sum }_{i=0}^{n-1}d(x_i,x_{i+1}),E(x):={\sum }_{i=0}^{n-1}d(x_i,x_{i+1}{)}^2\mathrm{.}$ The mesh-$h$ polygon space is $P_h(n):=\{x\in X^n:\mathrm{mesh}(x)\le h\}$; its points are called cyclic small-mesh polygons. Fixing $n$ (rather than letting the vertex count grow) is essential: the quantitative constants of this page depend on $n$ but not on $h$, and they are not uniform in a variable vertex count.

(3) The midpoint operation. For $x\in X^n$ with $\mathrm{mesh}(x)<l$, consecutive vertices satisfy $d(x_i,x_{i+1})<l<\pi \mathrm{/}2$, and the closed ball $\bar{B}(x_i,l)$ is CAT(1); by the convexity and uniqueness clause for balls of radius $<\pi \mathrm{/}2$ (Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences) the geodesic segment $[x_i,x_{i+1}]$ inside that ball is unique (Geodesics and geodesic metric spaces) and has a unique midpoint. Define $f(x)\in X^n$ by $f(x{)}_i:=\mathrm{mid}(x_i,x_{i+1})$ for $0\le i<n$. Continuity of $f$ on $\{\mathrm{mesh}<l\}$ and the invariance $\mathrm{mesh}(fx)\le \mathrm{mesh}(x)$, which make $f$ a self-map of $P_h(n)$, are proved in Existence of the uniform radius, continuity of the midpoint operation, the energy drop, its equality case, and convergence of zero-limit polygons ↗. Write $f^k$ for the $k$-fold iterate.

(4) The zero-limit basin. $C_h^0(n):=\{x\in P_h(n):f^k(x)\text{ is defined for every }k\ge 0\text{ and }{\lim }_{k\to \mathrm{\infty }}L(f^{k}x)=0\}$ (Limits and Cauchy sequences of reals). By (3) the definability condition is automatic on $P_h(n)$, so $C_h^0(n)=\{x\in P_h(n):{\lim }_{k}L(f^{k}x)=0\}$; this set is the zero-limit basin. It is defined purely through the iterated lengths, without reference to short-loop homotopy; its identification with a short-loop class is proved later on this page, after its topological properties are established.

(5) Conventions. A tuple is constant if all its entries are equal; a constant tuple has $\mathrm{mesh}=L=E=0$ and is fixed by $f$ (the midpoint of a degenerate segment is its point, in accordance with Geodesics and geodesic metric spaces). If some consecutive vertices coincide while the tuple is not constant, the corresponding edge contributes $0$ to mesh, length and energy, and the midpoint operation is applied to the (possibly degenerate) pair by the same rule.