Idempotent and multivariate copulas with fractal support

Using special iterated function systems (IFS) Fredricks et al. (2005) constructed two-dimensional copulas with fractal supports and showed that for every s ∈ ( 1,2 ) there exists a copula A whose support has Hausdorff dimension s. In the current paper we present a stronger version and prove that the same result holds for the subclass of idempotent copulas. Additionally we show that every doubly stochastic idempotent matrix N (having neither minimum nor maximum rank) induces a family of idempotent copulas such that, firstly, the corresponding Markov kernels transform according to N and, secondly, the set of Hausdorff dimensions of the supports of elements of the family covers (1,2). Furthermore we generalize the IFS approach to arbitrary dimensions d ≥ 2 and show that for every s ∈ ( 1 , d ) we can find a d-dimensional copula whose support has Hausdorff dimension s.