Continuous functions on Hausdorff continua

We consider the question: given a set X   and a function T:X→XT:X→X, does there exist a topology on X with respect to which X is a Hausdorff continuum and T is continuous? We answer the question for the class of κ  -to-one surjective functions (κ≥cκ≥c) with finitely based orbit spectrums. We show that if a member T:X→XT:X→X of the class is compactifiable (X can be equipped with a compact Hausdorff topology with respect to which T continuous), then there is a Hausdorff continuum Y   and a continuous function T′:Y→YT′:Y→Y with the same orbit spectrum as that of T.