Quasisymmetric map

Remarks

Example: an $L$-bi-Lipschitz map, with $L^{-1} d_X(x, y) \le d_Y(f(x), f(y)) \le L\, d_X(x, y)$, is $\eta$-quasisymmetric with $\eta(t) = L^2 t$. Indeed the numerator is at most $L\, d_X(x, a)$ and the denominator is at least $L^{-1} d_X(x, b)$. A similarity is quasisymmetric with $\eta(t) = t$.

Non-example: $f(x) = e^x$ on $\R$ is not quasisymmetric. For $s > 0$ take $a = x + s$ and $b = x - s$; the ratio on the right is $1$, while the ratio on the left is $\frac{e^s - 1}{1 - e^{-s}} = e^s$. No single value $\eta(1)$ can bound $e^s$ for all $s$.

A quasisymmetric map is a topological embedding, and its inverse on $f(X)$ is quasisymmetric too. An increasing homeomorphism $h$ of $\R$ is quasisymmetric if and only if there is $M \ge 1$ with $\frac{1}{M} \le \frac{h(x + t) - h(x)}{h(x) - h(x - t)} \le M$ for all $x \in \R$ and $t > 0$.