Multifractal lattice and group theory

The multifractal lattice Qmf is an object defined on a square using a section parameter ζ. Qmf has been used to study percolation in heterogeneous multifractal structures. In this work we use a group theory approach to explore mathematical properties of Qmf. The self-affine object Qmf is described by the combination of distinct discrete groups: the finite groups of rotation and inversion and the infinite groups of translation and dilation. We address the cell elements of the lattice Qmf using a Cayley tree. We determine the Cartesian coordinates of each cell using group properties in a recursive equation. The rich group structure of Qmf allows an infinite number of distinct tilling for a single ζ.