Let
![I = (0, 1]](/media/m/b/3/1/b31662c00b12d2c1bafe576a0d350e1d.png)
be the unit interval of the real line. For a given number
![a \in (0, 1)](/media/m/7/2/7/7275aba3432898ef4928002f3b402427.png)
we define a map
![T : I \to I](/media/m/d/2/7/d27f8ae027990b164668daa44386a01b.png)
by the formula
if
{{ INVALID LATEX }}
Show that for every interval
![J \subset I](/media/m/b/f/7/bf776acd44798eb09e3ccbcff569b418.png)
there exists an integer
![n > 0](/media/m/4/6/e/46ec46da606ada2d1f5ac481e6ae36f9.png)
such that
%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 $T^n(J) \cap J \neq \emptyset.$