  In the dynamics of a rotation of the unit circle by an irrational angle
$\alpha\in(0,1)$, we study the evolution of partitions whose atoms are finite
unions of left-closed right-open intervals with endpoints lying on the past
trajectory of the point $0$. Unlike the standard framework, we focus on
partitions whose atoms are disconnected sets. We show that the refinements of
these partitions eventually coincide with the refinements of a preimage of the
Sturmian partition, which consists of two intervals $[0,1-\alpha)$ and
$[1-\alpha,1)$. In particular, the refinements of the partitions eventually
consist of connected sets, i.e., intervals. We reformulate this result in terms
of Sturmian subshifts: we show that for every non-trivial factor mapping from a
one-sided Sturmian subshift, satisfying a mild technical assumption, the
sliding block code of sufficiently large length induced by the mapping is
injective.
