On the integrability of the transfer dynamics of non-involutive Yang-Baxter maps

It is well known that, given a Yang-Baxter map, there is a hierarchy of commuting transfer maps, which arise out of the consideration of initial value problems. In this paper, we show that one can construct invariants of the transfer maps corresponding to the $n$-periodic initial value problem on the two-dimensional lattice, using the same generating function that is used to produce invariants of the Yang-Baxter map itself. Moreover, we discuss the Liouville integrability of these transfer maps. Finally, we consider four-dimensional Yang-Baxter maps corresponding to the nonlinear Schr\"odinger (NLS) equation and the derivative nonlinear Schr\"odinger (DNLS) equation which have recently appeared. We show that the associated transfer maps are completely integrable.