Školjka

%V0 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 $$T (x, y) =\left\{\begin{array}{cc}x+(1-a),&\mbox{ if }0< x\leq a,\\ \text{ }\\ x-a, &\mbox{ if }a < x\leq 1.\end{array}\right$$ Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap J \neq \emptyset.$