Independence-preserving transformations are central in classical characterizations such as Lukacs’ theorem and the Matsumoto–Yor property. In this talk, the emphasis is on a more recent source of such phenomena: quadrirational Yang–Baxter maps (of Adler–Bobenko–Suris type [2:2]), viewed as real-analytic involutions on $(0,\infty)^2$.
Given a diffeomorphism $F$ on a product domain, the requirement that independent inputs $(X,Y)$ yield independent outputs $(U,V)=F(X,Y)$ for some product law is equivalent to a separated functional equation for the logarithms of the densities. The key point is that with this point of view, the characterisation of all independence-preserving product measures reduces to describing the Abelian relations of the associated planar 4-web given by the foliations $(x, y, u, v)$. Using the Bol bound for planar 4-webs, one obtains that for each of the three quadrirational involutions $H_I^+$, $H_{II}^+$ and $H_{III}^A$, the space of Abelian relations has dimension 3, and explicit bases can be written down. This yields a complete three-parameter family of independence-preserving product distributions, which is beta-prime, Kummer and GIG-type families.