  In our earlier paper [A square root map on Sturmian words, Electron. J.
Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal
squareful infinite word $s$ contains exactly six minimal squares and can be
written as a product of these squares: $s = X_1^2 X_2^2 \cdots$. The square
root $\sqrt{s}$ of $s$ is the infinite word $X_1 X_2 \cdots$ obtained by
deleting half of each square. We proved that the square root map preserves the
languages of Sturmian words (which are optimal squareful words). The dynamics
of the square root map on a Sturmian subshift are well understood. In our
earlier work, we introduced another type of subshift of optimal squareful words
which together with the square root map form a dynamical system. In this paper,
we study these dynamical systems in more detail and compare their properties to
the Sturmian case. The main results are characterizations of periodic points
and the limit set. The results show that while there is some similarity it is
possible for the square root map to exhibit quite different behavior compared
to the Sturmian case.
