Investigating the level sets \(\varvec{L^t}\) of Archimax copulas \(\varvec{C \in \mathcal {C}_{am}}\) , we establish that these sets can be characterized in terms of certain convex functions \(\varvec{f^s}\) and non-decreasing functions \(\varvec{g^t}\) . Motivated by the results in Mai and Scherer (Extremes 14, 311-324 2011) and Trutschnig et al. (Extremes 19, 405-427 2016), which examine the way bivariate Extreme Value copulas distribute their mass, we extend these findings to the larger family of bivariate Archimax copulas \(\varvec{{C}_{am}}\) . Working with Markov kernels (conditional distributions), we analyze the mass distributions of Archimax copulas \(\varvec{C \in \mathcal {C}_{am}}\) and show that the support of \(\varvec{C}\) is determined by some functions \(\varvec{f^0}\) , \(\varvec{g^L}\) , and \(\varvec{g^R}\) . Additionally, we prove that the discrete component (if any) of \(\varvec{C}\) concentrates its mass on the graphs of the afore-mentioned functions \(\varvec{f^s}\) or \(\varvec{g^t}\) . Recognizing the close relationship between the level sets \(\varvec{L^t}\) of a copula \(\varvec{C}\) and its Kendall distribution function \(\varvec{F_C^K}\) , we provide an alternative proof for the representation of \(\varvec{F_C^K}\) for arbitrary Archimax copulas \(\varvec{C \in \mathcal {C}_{am}}\) and derive simple expressions for the level set masses \(\varvec{\mu _C(L^t)}\) . Building upon the fact that Archimax copulas \(\varvec{C \in \mathcal {C}_{am}}\) can be represented via two univariate probability measures \(\boldsymbol\gamma\) and \(\varvec{\vartheta }\) — so-called Williamson and Pickands dependence measures — we show that absolute continuity, discreteness, and singularity properties of these measures \(\varvec{\gamma }\) and \(\varvec{\vartheta }\) carry over to the corresponding Archimax copula \(\varvec{C_{\gamma , \vartheta }}\) . Finally, we derive conditions on \(\varvec{\gamma }\) and \(\varvec{\vartheta }\) such that the support of the absolutely continuous, discrete, or singular component of \(\varvec{C_{\gamma , \vartheta }}\) coincides with the support of \(\varvec{C_{\gamma , \vartheta }}\) .
