  We look at the properties of clusters of order parameter at critical points
in thermal systems and consider their significance to statistical-mechanical
ground rules. These properties have been previously obtained through the
saddle-point approximation in a coarse-grained partition function. We examine
both static and dynamical aspects of a single large cluster and indicate that
these properties fall outside the canonical Boltzmann-Gibbs (BG) scheme.
Specifically: 1) The faster than exponential growth with cluster size of the
space-integrated order parameter suggests nonextensivity of the BG entropy but
extensivity of a q-entropy expression. 2) The finding that the time evolution
of the order parameter is described by the dynamics of an intermittent
nonlinear map implies an atypical sensitivity to initial conditions compatible
with q-statistics and displays an 'aging' scaling property. 3) Both, the
approach to criticality and the infinite-size cluster limit at criticality
manifest through a crossover from canonical to q-statistics and we discuss the
nonuniform convergence associated to these features. Key words: critical
clusters, saddle-point approximation, q-entropy, intermittency, aging
