  While structures and bifurcations controlling tracer particle transport and
mixing have been studied extensively for systems with only
stretching-and-folding, and to a lesser extent for systems with only
cutting-and-shuffling, few studies have considered systems with a combination
of both. We demonstrate two bifurcations for non-mixing islands associated with
elliptic periodic points that only occur in systems with combined
cutting-and-shuffling and stretching-and-folding, using as an example a map
approximating biaxial rotation of a less-than-half-full spherical granular
tumbler. First, we characterize a bifurcation of elliptic island containment,
from containment by manifolds associated with hyperbolic periodic points to
containment by cutting line tangency. As a result, the maximum size of the
non-mixing region occurs when the island is at the bifurcation point. We also
demonstrate a bifurcation where periodic points are annihilated by the
cutting-and-shuffling action. Chains of elliptic and hyperbolic periodic points
that arise when invariant tori surrounding an elliptic point break up
[according to Kolmogorov-Arnold-Moser (KAM) theory] can annihilate when they
meet a cutting line. Consequently, the Poincar\'{e} index (a topological
invariant of smooth systems) is not preserved.
