ILL 1976 Problem 29

Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula if $$T (x, y) = \begin{cases} x + (1 - a),&\text{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \text{ if } a < x \leq 1.\end{cases}$$

Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$