Nonsimultaneous hierarchy: multifractal analysis

Abstract

We define a simple hierarchical model in two dimensions incorporating two independent and nonsimultaneous rescaling procedures in orthogonal directions. The multifractal framework and corresponding thermodynamic formalism necessary for analyzing models of this type are set up using the concept of partial dimensions, and the model is then analyzed. We define an evolution space such that distinct paths in the evolution space correspond to distinct ways of constructing the hierarchy. If we utilize four scale Cantor sets in the construction of our hierarchy the equivalent Hamiltonian corresponds to that due to nearest-neighbor interactions between spins at the sites of a one-dimensional path on a two-dimensional lattice and is path dependent.