North–South Homeomorphisms of the Sierpiński Carpet and the Menger Curve

A homeomorphism $f$ is North–South (or loxodromic) if it has an attracting fixed point $x^+$, a repelling fixed point $x^-$, and $\lim_{n\to+\infty} f^{\pm n}(x)=x^\pm$ for every $x\neq x^+,x^-$. We show that, up to conjugacy, there are exactly four North–South homeomorphisms on the Sierpiński curve $X$, and one on the Menger curve $M$. Every countable group acts effectively on the Menger curve $M$ (but there exist many finite groups with no effective action on the Sierpiński curve). All epimorphisms from $\pi_1M$ to $\mathbb Z$ are equivalent (up to a homeomorphism of $M$); the analogous statement for $\mathbb Z/2\mathbb Z$ is false.