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 ${\mathcal{S}}_{\mathit{T}}$ induced by pairs $\mathit{T}=(T_1\times T_2)$ of measure preserving transformations on $([0,1],\mathcal{B}([0,1]),\lambda )$ on the family $\mathcal{C}$ 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 $\mathcal{C}$ with respect to the uniform metric $d_{\infty }$. After deriving some general properties of the set ${\mathrm{\Omega }}_{\mathit{T}}$ of all ${\mathcal{S}}_{\mathit{T}}$-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 ${\mathrm{\Omega }}_{\mathit{T}}$. 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 ${\mathrm{\Omega }}_{\mathit{T}}$, including the surprising fact that ${\mathrm{\Omega }}_{\mathit{T}}$ 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.

考虑在x和y方向上众所周知的混洗操作会产生所谓的双变量 copula 的双重混洗。我们研究了双洗牌算子的连续性${\mathcal{秒}}_{\mathit{吨}}$ 成对诱发 $\mathit{吨}=({吨}_1\times {吨}_2)$ 测量保持变换 $([0,1],\mathcal{乙}([0,1]),\lambda )$ 关于家庭 $\mathcal{C}$所有的二元Copula函数的，分析其相互关系与明星/马尔科夫的产品，并显示每个左手和各右可逆系词一个集合的所有可能的双重洗牌的一个是在茂密$\mathcal{C}$ 关于统一度量 $d_{\infty }$. 在导出集合的一些一般属性之后${\mathrm{\Omega }}_{\mathit{吨}}$ 所有的 ${\mathcal{秒}}_{\mathit{吨}}$-不变的copulas我们关注的情况是 ${吨}_1,{吨}_2$ 强烈混合并表明在这种情况下，乘积 copula Π 是 ${\mathrm{\Omega }}_{\mathit{吨}}$. 此外，受 Horanská 和 Sarkoci 近期论文（Fuzzy Sets and Systems 378, 2018）的启发，我们研究了由所谓的 Lüroth 映射对引起的双重洗牌，并推导出${\mathrm{\Omega }}_{\mathit{吨}}$，包括令人惊讶的事实 ${\mathrm{\Omega }}_{\mathit{吨}}$ 包含无数极值点，它们（解释为双重随机测度）彼此成对相互奇异，并且允许显式构造。