  We investigate the problem of when big mapping class groups are generated by
involutions. Restricting our attention to the class of self-similar surfaces,
which are surfaces with self-similar ends space, as defined by Mann and Rafi,
and with 0 or infinite genus, we show that, when the set of maximal ends is
infinite, then the mapping class groups of these surfaces are generated by
involutions, normally generated by a single involution, and uniformly perfect.
In fact, we derive this statement as a corollary of the corresponding statement
for the homeomorphism groups of these surfaces. On the other hand, among
self-similar surfaces with one maximal end, we produce infinitely many examples
in which their big mapping class groups are neither perfect nor generated by
torsion elements. These groups also do not have the automatic continuity
property.
