Geodesic
Periodic trajectories in a~Denjoy counterexample
Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 617-620.

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It is shown that for the parametric class of piecewise linear maps $$ f(x)=\begin{cases} \max(k_1x+1,w), 0, \\ \min(k_2x-1,w), \geq0 \end{cases} $$ ($k_1$ 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. 
@article{FPM_2000_6_2_a17,
     author = {L. K. Bakalinski},
     title = {Periodic trajectories in {a~Denjoy} counterexample},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {617--620},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2000_6_2_a17/}
}
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L. K. Bakalinski. Periodic trajectories in a~Denjoy counterexample. Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 2, pp. 617-620. http://geodesic.mathdoc.fr/item/FPM_2000_6_2_a17/