Diluted Ising model with competing interactions

We introduce aperiodic, but deterministic, dilution of bonds in the Ising model with competing ferro and antiferromagnetic interactions between first and second neighbors along the branches of a Cayley tree. The problem is formulated as a non-linear dissipative map, whose attractors correspond to solutions deep inside the tree. We use a scheme of successive periodic approximations to obtain the main modulated structures of the phase diagrams. The paramagnetic lines, as well as some other features of the phase diagrams, can be obtained from closed expressions.