ILL 1976 Problem 28

Let $Q$ be a unit square in the plane: $Q = [0, 1] \times [0, 1]$ . Let $T :Q \longrightarrow Q$ be defined as follows: $$T(x, y) =\begin{cases} (2x, \frac{y}{2}) &\text{ if } 0 \le x \le \frac{1}{2};(2x - 1, \frac{y}{2}+ \frac{1}{2})&\text{ if } \frac{1}{2} < x \le 1.\end{cases}$$ Show that for every disk $D \subset Q$ there exists an integer $n > 0$ such that $T^n(D) \cap D \neq \emptyset.$