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On the non-autonomous form of the Q4 mapping and its relation to elliptic Painlevé equations

We apply the deautonomization procedure to mappings obtained by one-dimensional reductions of the Q4 and Q3 integrable lattice systems of Adler, Bobenko and Suris. We show that in the case of the Q4 mapping the non-autonomous forms are elliptic discrete Painlevé equations while for Q3 the deautonomization leads to linearizable systems.