Geodesic
Externally periodic automata
Diskretnaya Matematika, Tome 17 (2005) no. 1, pp. 68-72.

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We describe the automata which transfer any periodic input sequence into a periodic output sequence. We suggest an algorithm for checking whether a given automaton has this property and give upper bounds for the complexity of the algorithm. 
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     author = {A. V. Babash},
     title = {Externally periodic automata},
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A. V. Babash. Externally periodic automata. Diskretnaya Matematika, Tome 17 (2005) no. 1, pp. 68-72. http://geodesic.mathdoc.fr/item/DM_2005_17_1_a5/

[1] Kobrinskii N. E., Trakhtenbrot B. A., Vvedenie v teoriyu konechnykh avtomatov, Nauka, Moskva, 1962