Geodesic
North--South Homeomorphisms of the Sierpi\'nski Carpet and the Menger Curve
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 305-311.

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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. 
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     title = {North--South {Homeomorphisms} of the {Sierpi\'nski} {Carpet} and the {Menger} {Curve}},
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G. Levitt. North--South Homeomorphisms of the Sierpi\'nski Carpet and the Menger Curve. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems and related problems of geometry, Tome 244 (2004), pp. 305-311. http://geodesic.mathdoc.fr/item/TM_2004_244_a12/

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