  We report detailed investigation of the existence and stability of mixed and
demixed modes in binary atomic Bose-Einstein condensates with repulsive
interactions in a ring-trap geometry. The stability of such states is examined
through eigenvalue spectra for small perturbations, produced by the
Bogoliubov-de Gennes equations, and directly verified by simulations based on
the coupled Gross-Pitaevskii equations, varying inter- and intra-species
scattering lengths so as to probe the entire range of miscibility-immiscibility
transitions. In the limit of the one-dimensional (1D) ring, i.e., a very narrow
one, stability of mixed states is studied analytically, including
hidden-vorticity (HV) modes, i.e., those with opposite vorticities of the two
components and zero total angular momentum. The consideration of demixed 1D
states reveals, in addition to stable composite single-peak structures, double-
and triple-peak ones, above a certain particle-number threshold. In the 2D
annular geometry, stable demixed states exist both in radial and azimuthal
configurations. We find that stable radially-demixed states can carry arbitrary
vorticity and, counter-intuitively, the increase of the vorticity enhances
stability of such states, while unstable ones evolve into randomly oscillating
angular demixed modes. The consideration of HV states in the 2D geometry
expands the stability range of radially-demixed states.
