Some results on homeomorphisms between fractal supports of copulas

We consider parametric classes (Tr)r∈(0,1/2)(Tr)r∈(0,1/2) of so-called transformation matrices and their induced families (Ar)r∈(0,1/2)(Ar)r∈(0,1/2) and (μr)r∈(0,1/2)(μr)r∈(0,1/2) of two-dimensional copulas and doubly stochastic measures with fractal support respectively. By using tools from Symbolic Dynamics we show that for each pair r,r′∈(0,1/2)r,r′∈(0,1/2) with r≠r′r≠r′ there exists a homeomorphism Hrr′Hrr′ between the supports of μrμr and μr′μr′ mapping a Borel set of μrμr-measure one to a set of μr′μr′-measure zero. Differentiability properties of these homeomorphisms are studied and Hausdorff dimensions of related sets are calculated. Several examples and graphics illustrate the main results.