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

Abstract

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 $H_{\mathrm{I}}$, $H_{\mathrm{II}}$ and $H_{\mathrm{III}}^A$ 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 $H_{\mathrm{I}}$, $H_{\mathrm{II}}$ and $H_{\mathrm{III}}^A$ 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 Painlevé equations.

References in corpus (4)

• Yang-Baxter maps: dynamical point of view
• Simplex and Polygon Equations
• Relativistic Collisions as Yang-Baxter maps
• On a class of third order mappings with two rational invariants