text: abstract
  a copula of continuous random variables x and y is called an implicit
  dependence copula if there exist functions α and β such that α(x) =
  β(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 σalgebras. 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.