  Expanding upon our previous study of competing critical phenomena in black
hole formation, we numerically investigate the behavior of dominant exponents
across the boundary separating asymptotically dispersing and collapsing regions
in a two-dimensional configuration space of initial data. We find that across
the Type II boundary section the dominant exponent remains constant, equal to
the reciprocal of Choptuik's well-known quasi-universal value, whereas across
the Type I section the exponent noticeably varies. We postulate that this
change reflects the existence of a third critical solution in addition to the
two primary competing solutions, possibly another member of the family of
metastable soliton stars constituting the Type I attractor.
