  We consider reachability in dynamical systems with discrete linear updates,
but with fixed digital precision, i.e., such that values of the system are
rounded at each step. Given a matrix $M \in \mathbb{Q}^{d \times d}$, an
initial vector $x\in\mathbb{Q}^{d}$, a granularity $g\in \mathbb{Q}_+$ and a
rounding operation $[\cdot]$ projecting a vector of $\mathbb{Q}^{d}$ onto
another vector whose every entry is a multiple of $g$, we are interested in the
behaviour of the orbit $\mathcal{O}={<}[x], [M[x]],[M[M[x]]],\dots{>}$, i.e.,
the trajectory of a linear dynamical system in which the state is rounded after
each step. For arbitrary rounding functions with bounded effect, we show that
the complexity of deciding point-to-point reachability---whether a given target
$y \in\mathbb{Q}^{d}$ belongs to $\mathcal{O}$---is PSPACE-complete for
hyperbolic systems (when no eigenvalue of $M$ has modulus one). We also
establish decidability without any restrictions on eigenvalues for several
natural classes of rounding functions.
