IMO 1976 SL 7

Problem

(POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given number a \in(0, 1) we define a map T : I \toI by the formula T (x, y) = . x + (1 −a) if 0 < x \leqa, x −a if a < x \leq1. Show that for every interval J \subsetI there exists an integer n > 0 such that T n(J) \capJ ̸= \emptyset.

Solution

The map T transforms the interval (0, a] onto (1 −a, 1] and the interval (a, 1] onto (0, 1 −a]. Clearly T preserves the measure. Since the measure of the interval [0, 1] is finite, there exist two positive integers k, l > k such that T k(J) and T l(J) are not disjoint. But the map T is bijective; hence T l−k(J) and J are not disjoint.