Periodic trajectories in a Denjoy counterexample

It is shown that for the parametric class of piecewise linear maps
$$ f(x)=\begin{cases} \max(k_1x+1,w), &x<0, \\ \min(k_2x-1,w), &x\geq0 \end{cases} $$
($k_1$ and $k_2$ are greater than one) the range of the parameter $w$, where iterations $x_{n+1}=f(x_n)$ are nonperiodic, has zero Lebesgue measure.