We discuss the question of extending homeomorphisms between closed subsets of the Cantor cube $D^{\tau }$. It is established that any homeomorphism between two closed negligible subsets of $D^{\tau }$ can be extended to an autohomeomorphism of $D^{\tau }$.

Let G be the group $\text {PAff}_{+}(\mathbb R/\mathbb Z)$ of piecewise affine circle homeomorphisms or the group ${\operatorname {\mathrm {Diff}}}^{{\kern1pt}\infty }(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.

We study the hyperspace dynamics induced fromgeneric continuous maps and fromgeneric homeomorphisms of the Cantor space, with emphasis on the notions of Li– Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing, and recurrence.

Given an integer $n\,\ge \,3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f:\,X\,\to \,X$, it is true that for $\mu$-almost every point $x$ in $X$ the restriction of $f$ (respectively of ${{f}^{-1}}$) to the omega limit set $\omega \left( f,\,x \right)$ (respectively to the alpha limit set $\alpha \left( f,\,x \right)$) is topologically conjugate to the universal odometer.

Given an integer $n\,\ge \,3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu $ on $X$, we prove that for the typical homeomorphism $f\,:\,X\,\to \,X$, it is true that for $\mu $-almost every point $x$ in $X$ the limit set $\omega (f,\,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega (f,\,x)$ has a dense orbit in $\omega (f,\,x)$, $f$ is non-sensitive at each point of $\omega (f,\,x)$, and the function $a\,\to \,\omega (f,\,a)$ is continuous at $x$.