  A copula of continuous random variables $X$ and $Y$ is called an
\emph{implicit dependence copula} if there exist functions $\alpha$ and $\beta$
such that $\alpha(X) = \beta(Y)$ almost surely, which is equivalent to $C$
being factorizable as the $*$-product of a left invertible copula and a right
invertible copula. Every implicit dependence copula is supported on the graph
of $f(x) = g(y)$ for some measure-preserving functions $f$ and $g$ but the
converse is not true in general.
  We obtain a characterization of copulas with implicit dependence supports in
terms of the non-atomicity of two newly defined associated $\sigma$-algebras.
As an application, we give a broad sufficient condition under which a
self-similar copula has an implicit dependence support. Under certain extra
conditions, we explicitly compute the left invertible and right invertible
factors of the self-similar copula.
