Geodesic
Periodically externally hereditary permutation automata
Diskretnaya Matematika, Tome 17 (2005) no. 3, pp. 68-79

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We suggest an algorithm which checks the following property of a given finite permutational automaton: for any periodic input sequence and any initial state the period of the corresponding output sequence is divided by the period of the input sequence. Estimates of the complexity of such algorithms are given. 
@article{DM_2005_17_3_a6,
     author = {A. V. Babash},
     title = {Periodically externally hereditary permutation automata},
     journal = {Diskretnaya Matematika},
     pages = {68--79},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2005_17_3_a6/}
}
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A. V. Babash. Periodically externally hereditary permutation automata. Diskretnaya Matematika, Tome 17 (2005) no. 3, pp. 68-79. http://geodesic.mathdoc.fr/item/DM_2005_17_3_a6/