Aperiodic order in dimension-drop parameter sets

Determine for which natural classes of self-similar sets the parameters exhibiting dimension drop are organised by an underlying dynamical or combinatorial order, and establish whether such parameter sets can possess substitution structure related systematically to the algebraic mechanisms responsible for dimension drop.

Background

The paper studies the family of self-similar sets E(x,y) generated by the four affine maps with digits 0, x, y, and x+y and contraction ratio 1/4. For natural parameters, the authors completely classify the topological type of E(x,y) and show that the associated dimension-drop parameters form a two-colour tiling of the lattice N².

Although this tiling is shown to be aperiodic and to arise as a factor of a substitution tiling on four arithmetic states, the authors leave unresolved whether analogous dynamical or combinatorial organisation occurs for broader natural classes of self-similar sets. They also ask whether substitution structures can occur systematically and whether such structures can be connected to the algebraic causes of dimension drop.