Fixed point sets of Tychonov cubes

We give a spectral characterization of the compacta (compact Hausdorff spaces) which admit embeddings into uncountable products of the closed unit interval such that the embedded images coincide with the fixed point sets of (continuous) self-mappings of those products. An example of P. Koszmider is used to show that there is a zero-dimensional compactum of weight ω1 which admits no such embedding into any ANR(compact Hausdorff)-spaces. This is in contrast to the metric case where it is not known if every nonempty closed subset of an ANR-space X is the fixed point set of a self-mapping of X.