Question 7
(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 $T:I\to I$ by the formula
$$T(x,y)=\begin{cases}x+(1-a)&\text{if }0\lt x\le a,\\x-a&\text{if }a\lt x\le1.\end{cases}$$
Show that for every interval $J\subset I$ there exists an integer $n\gt 0$ such that $T^n(J)\cap J\ne\varnothing$.
$$T(x,y)=\begin{cases}x+(1-a)&\text{if }0\lt x\le a,\\x-a&\text{if }a\lt x\le1.\end{cases}$$
Show that for every interval $J\subset I$ there exists an integer $n\gt 0$ such that $T^n(J)\cap J\ne\varnothing$.