We investigate strictly non-relativistic thermal effects on the dispersion of lefthanded (LH) ion cyclotron waves (ICW's) with real frequency and complex wave vector, propagating parallel to a uniform ambient magnetic field. Changes to the topology of the cold-plasma dispersion relations in the vicinity of the ion gyrofrequencies are studied in plasmas consisting predominantly of protons with a small admixture of a heavy ion. The two branches of the LH mode reconnect near the heavy-ion gyrofrequency as the heavy-ion temperature is increased or its relative density is reduced. The reconnection results in one mode in which waves can propagate at all frequencies below the proton gyrofrequency and another which allows propagation only in a narrow frequency range extending upwards from the cut-off frequency to a regime where strong damping occurs. The topology of the reconnected dispersion curves is quite different from that seen in the real wave vector – complex frequency case. This work is relevant to theories of ion heating and acceleration in multi-ion plasmas as are found in the solar wind, in solar and stellar flares, and in the Earth's magnetosphere. In particular, strongly species-dependent heating and acceleration can arise from wave–particle interactions between the various ionic species, and ICW's at frequencies near the respective ion gyrofrequencies. These interactions depend critically on the wave dispersion.