anomalous region. In contrast, the bimodality of the imaginary branch is much more closely correlated with the region of the structural anomaly. The bimodal character of the both branches of the INM spectrum suggests that such core-softened fluids may show multiple time-scale behaviour similar to that seen in hydrogen-bonded systems. The Einstein frequency shows an essentially monotonic dependence on density along an isotherm. The temperature dependence of the Einstein frequency is weak and monotonically decreasing with temperature at low densities and non-monotonic at high temperatures. In contrast to the Einstein frequency, the fraction of imaginary frequencies shows very anomalous behaviour in comparison to simple liquids, with an extended density regime over which F$_{im}$ increases with increasing density. While the low density boundary of this region coincides with that of the structural anomaly, the high density boundary occurs at very high densities well beyond the structurally anomalous regime. Previous INM studies of liquids have largely connected the diffusivity with the fraction of imaginary modes. Our results show that information about diffusivity is largely contained in the behaviour of the imaginary branch of the INM spectra, but factors such as the bimodality of the frequency distribution in this branch must be considered in addition to F$_{im}$ . Given the validity of excess entropy scaling for the diffusivities, we introduce INM-based estimators of the entropy, to connect the energy landscape with liquid state thermodynam- ics and kinetics. The conceptually simplest INM-based estimator is to treat the liquid as a collection of 3-dimensional harmonic oscillators vibrating at a single frequency, correspond- ing to the Einstein frequency. The Einstein model entropy shows a very steep decrease with density along isotherms with a very weak signature of the onset of the structurally anomalous regime. An alternative approach to developing an INM-based estimator of the entropy that we have explored is to assume that the total entropy of the fluid can be written as a sum of contributions from 3 NF$_{r}$ harmonic modes and 3 NF$_{im}$ imaginary modes. The real branch of the INM spectrum can be used to estimate the harmonic contribution, s ∗ $_{r}$, exactly. The entropy contribution of the imaginary branch, s ∗ $_{im}$, is then given by the difference of the thermodynamic entropy, s , and the real branch contribution, s ∗ $_{r}$. The temperature and density dependence of s ∗ r carries virtually no signature of the liquid-state anomalies, and seems to reflect only the generic effects of thermal fluctuations and spatial confinement on the entropy In contrast, s ∗ im = s ∗ − s ∗ $_{r}$, captures the behaviour of the entropy within the 18