Title: Periods and factors of weak model sets,
Abstract: There is a renewed interest in weak model sets due to their connection to
$\mathcal B$-free systems, which emerged from Sarnak's program on the Möbius
disjointness conjecture. Here we continue our recent investigation
[arXiv:1511.06137] of the extended hull ${\mathcal M}^{\scriptscriptstyle
G}_{\scriptscriptstyle W}$, a dynamical system naturally associated to a weak
model set in an abelian group $G$ with relatively compact window $W$. For
windows having a nowhere dense boundary (this includes compact windows), we
identify the maximal equicontinuous factor of ${\mathcal M}^{\scriptscriptstyle
G}_{\scriptscriptstyle W}$ and give a sufficient condition when ${\mathcal
M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is an almost 1:1 extension of
its maximal equicontinuous factor. If the window is measurable with positive
Haar measure and is almost compact, then the system ${\mathcal
M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ equipped with its Mirsky
measure is isomorphic to its Kronecker factor. For general nontrivial ergodic
probability measures on ${\mathcal M}^{\scriptscriptstyle
G}_{\scriptscriptstyle W}$, we provide a kind of lower bound for the Kronecker
factor. All relevant factor systems are natural $G$-actions on quotient
subgroups of the torus underlying the weak model set. These are obtained by
factoring out suitable window periods. Our results are specialised to the usual
hull of the weak model set, and they are also interpreted for ${\mathcal
B}$-free systems.