  We consider impurity atoms embedded in a two-component Bose-Einstein
condensate in a quasi-one dimensional regime. We study the effects of repulsive
coupling between the impurities and Bose species on the equilibrium of the
system for both miscible and immiscible mixtures by numerically solving the
underlying coupled Gross-Pitaevskii equations. Our results reveal that the
presence of impurities may lead to a miscible-immiscible phase transition due
to the interaction of the impurities and the two condensates. Within the realm
of the Bogoliubov-de Gennes equations we calculate the quantum fluctuations due
to the different types of interactions. The breathing modes and the time
evolution of harmonically trapped impurities in both homogeneous and
inhomogeneous binary condensates are deeply discussed in the miscible case
using variational and numerical means. We show in particular that the
self-trapping, the miscibility and the inhomogeneity of the trapped Bose
mixture may strongly modify the low-lying excitations and the dynamical
properties of impurities. The presence of phonons in the homogeneous Bose
mixture gives rise to the damping of breathing oscillations of impurities
width.
