Stability of the ω-limit set for unimodal transformations

Consider the map~$T\mapsto\omega (T)$ on the collection of unimodal transformations, where $\omega (T)$ is the $\omega$-limit set of~$T$. The collection of unimodal transformations is endowed with the topology of uniform convergence, and the collection of subsets of~$[0,1]$ is endowed with the Hausdorff metric. Conditions on a unimodal transformation~$T$ implying the continuity of the map~$S\mapsto\omega (S)$ at~$T$ are investigated.