  One of the few accepted dynamical foundations of non-additive
"non-extensive") statistical mechanics is that the choice of the appropriate
entropy functional describing a system with many degrees of freedom should
reflect the rate of growth of its configuration or phase space volume. We
present an example of a group, as a metric space, that may be used as the phase
space of a system whose ergodic behavior is statistically described by the
recently proposed $\delta$-entropy. This entropy is a one-parameter variation
of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from
the power-law entropies that have been recently studied. We use the first
Grigorchuk group for our purposes. We comment on the connections of the above
construction with the conjectured evolution of the underlying system in phase
space.
