\[
    (x_{n+1}, y_{n+1}) = \left \{ \begin{array}{ll}
	(2x_{n}, ay_{n}) & \textrm{for } 0 \leq x_{n} \leq \frac{1}{2} \\
	(2x_{n} - 1, ay_{n}) + \frac{1}{2} & \textrm{for } \frac{1}{2} \leq x_{n} \leq 1
    \end{array} \right .
\]