Copulas and associated fractal sets

The problem of constructing copulas whose supports are fractals has been studied by Fredricks, Nelsen and Rodríguez-Lallena [G.A. Fredricks, R.B. Nelsen, J.A. Rodríguez-Lallena, Copulas with fractal supports, Insurance Math. Econom. 37 (1) (2005) 42–48]. In this paper we continue on the path traced by these authors. We provide different types of families of self-similar copulas using techniques from Probability and Ergodic Theory to give properties on subsets of their fractal supports. In particular, we give new examples for those copulas and we analyze related topics with mutual singularity of the associated measures, Hausdorff dimension, and the connectedness of their supports.