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Any homeomorphism $h$ of the dyadic solenoid $S$ is isotopic to the "affine" homeomorphism of the form $g:x\mapsto \pm(2^n x+b)$ for some $n\in\mathbb Z$ and some $b\in S$. … If $h$ preserves the path-connected component $X$ of the neutral element, then so does the affine homeomorphism $g$, which implies that $b\in X$. …

The complete Erdos space $\mathfrak E_c$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E_c$. Proof. … Using the argument of the proof of Theorem 1, we can construct a self-homeomorphism $h_1$ of $\mathfrak E_c$ and a self-homeomorphism $h_2$ of the Cantor cube $2^{\omega}$ such that each orbit of the …

In the simplest form the Z-set Unknotting Theorem (proved by Bestvina) says that a homeomorphism $h:A\to B$ between two $Z$-sets of the $n$-dimensional Menger cube $M$ extends to a homeomorphism of $M$ … In particular, any homeomorphism between closed nowhere dense subsets of the Cantor set $M$ extends to a homeomorphism of $M$. …