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.