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 HI, HII and HIIIA 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 HI, HII and HIIIA 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 F and the H lists of  Yang-Baxter maps can be considered as the (k-1)-th - 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 the  autonomous versions of some hierarchies of discrete Painlevé equations.

Speaker

Dr Pavlos Kassotakis obtained his PhD  at the Department of Applied Mathematics from the University of Leeds in 2006. In 2008 he worked as a Post-doctoral Research Associate at the University of Sydney, Australia. From 2009 to 2011 he was a  visiting lecturer at the Department of Mathematics and Statistics, University of Cyprus, before obtaininghis second Post-doctoral Research Associate position at the University of Sydney. From 2014 he has worked at the Department of Mathematics and Statistics, University of Cyprus. During the academic year 2016-17 he also served as a visiting lecturer at the Department of Mathematics at the University of the Aegean in Samos, Greece.

Dr Kassotakis’ research interests include discrete and continuous integrable systems, integrable partial difference  equations, integrable ordinary difference equations, discrete and continuous Painlevé equations. The main focus of his research are integrable maps -  Yang-Baxter maps, tetrahedron maps, Laurent maps and Laurent phenomenon for discrete equations.