Some properties of double shuffles of bivariate copulas and (extreme) copulas invariant with respect to Lüroth double shuffles

Considering the well-known shuffling operation in x- and in y-direction yields so-called double shuffles of bivariate copulas. We study continuity properties of the double shuffle operator S _T induced by pairs T=(T ₁×T ₂) of measure preserving transformations on ([0,1],B([0,1]),λ) on the family C of all bivariate copulas, analyze its interrelation with the star/Markov product, and show that for each left- and for each right-invertible copula A the set of all possible double shuffles of A is dense in C with respect to the uniform metric d _∞. After deriving some general properties of the set Ω _T of all S _T-invariant copulas we focus on the situation where T ₁,T ₂ are strongly mixing and show that in this case the product copula Π is an extreme point of Ω _T. Moreover, motivated by a recent paper by Horanská and Sarkoci (Fuzzy Sets and Systems 378, 2018) we then study double shuffles induced by pairs of so-called Lüroth maps and derive various additional properties of Ω _T, including the surprising fact that Ω _T contains uncountably many extreme points which (interpreted as doubly stochastic measures) are pairwise mutually singular with respect to each other and which allow for an explicit construction.