Measure of noninvertibility of minimal maps

Minimal maps in compact metric spaces are known to be almost one-to-one. Thus, the set of points with more than one preimage is of first category. In the present paper we study the measure of this set with respect to the invariant measures of the considered minimal map. Among others, we give an example of a minimal self-mapping of a continuum such that the set of points with more than one preimage has positive measure for every invariant measure.