MathDB

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),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ 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.

7

Problems(1)