Given an integer $n \geq 3$, 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 $\omega(f,x)$ (respectively to the alpha limit set $\alpha(f,x)$) is topologically conjugate to the universal odometer.

Given an integer $n \geq 3$, 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 $f \colon X \to X$, 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 $x$.

Let $|K|$ be the metric polyhedron of a simplicial complex $K$. In this paper, we characterize a simplicial subdivision $K'$ of $K$ preserving the metric topology for $|K|$ as the one such that the set $K'{}^{(0)}$ of vertices of $K'$ is discrete in $|K|$. We also prove that two such subdivisions of $K$ have such a common subdivision.

We characterize maps between $n$-dimensional Nöbeling manifolds that can be approximated by homeomorphisms.

We obtain a homotopy splitting of the forget control map for controlled homeomorphisms over closed manifolds of nonpositive curvature.