Research Announcement SETS OF DISTANCES AND MAPPINGS

Given a function f defined on a metric space X, we denote by 6 the set of distancesf 6 = {dist(x,x'): x,x ' E X,f(x) = f(x l)}. f By a continuum we mean a connected compact metric space. If f: X + Y is a continuous mapping of a compact metric space X which lowers the dimension of X, then there exists a point Yo E Y such that the set f-1 (yO) is positive-dimensional. Thus in this case, f-l(yO) contains a non-degenerate continuum, and 6 contains an interval [0, a], where a> O. The latter propertyf can, however, be possessed also by those mappings which do not necessarily lower the dimension of the space. Then, in most cases, the distances of 6 filling up an interval may have tof selected between pairs of points belonging to many sets f-l(y) instead of just one. For example, it is known [6] that i