Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the H-I, H-II and H-III(A) Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the H-I, H-II and H-III(A) Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole F and H-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the H-list of Yang-Baxter maps can be considered as the (k - 1)-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to k-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painleve equations.