Critical behavior of the Ising model on a hierarchical lattice with aperiodic interactions

We write the exact renormalization-group recursion relations for nearest-neighbor ferromagnetic Ising models on Migdal–Kadanoff hierarchical lattices with a distribution of aperiodic exchange interactions according to a class of substitutional sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, as in the case of the Rudin–Shapiro sequence, the uniform fixed point in the parameter space cannot be reached from any physical initial conditions. We derive a criterion to check the relevance of the geometric fluctuations.