THE DYNAMICS OF A TYPICAL MEASURABLE FUNCTION ARE DETERMINED ON A ZERO MEASURE SET

Let A be a zero measure dense G(delta) subset of I = [0, 1], with M the set of measurable self-maps of I. There exists a residual set R subset of M such that for each f in R, the range of f is contained in A, and the function f is one-to-one. Moreover, there exists h : I -> I a Baire-2 function such that f (x) = h(x) a.e., and for any x is an element of I, the trajectory tau(x, h) is infinity-adic, so that the omega-limit set omega(x, h) is a Cantor set. Since the range of f is contained in A, it follows that for any x in I, there exists y in A such that the trajectory tau(f(x), f) = tau(y, f) subset of A. Speaking loosely, the dynamical structures of f are completely determined by its behavior on the set A.