  Systems of globally coupled logistic maps (GCLM) can display complex
collective behaviour characterized by the formation of synchronous clusters. In
the dynamical clustering regime, such systems possess a large number of
coexisting attractors and might be viewed as dynamical glasses. Glass
properties of GCLM in the thermodynamical limit of large system sizes $N$ are
investigated. Replicas, representing orbits that start from various initial
conditions, are introduced and distributions of their overlaps are numerically
determined. We show that for fixed-field ensembles of initial conditions, as
used in previous numerical studies, all attractors of the system become
identical in the thermodynamical limit up to variations of order $1/\sqrt{N}$
because the initial value of the coupling field is characterized by vanishing
fluctuations, and thus replica symmetry is recovered for $N\to \infty $. In
contrast to this, when random-field ensembles of initial conditions are chosen,
replica symmetry remains broken in the thermodynamical limit.
