Consensus of Julia Sets of Potts Models on Diamond-Like Hierarchical Lattice

The limit sets of zeros of the partition function for $\lambda $ -state Potts models on diamond-like hierarchical lattice are the Julia sets of functions in a family of rational functions. In this paper, the consensus problem of Julia sets generated by $\lambda $ -state Potts models on diamond-like hierarchical lattice is studied. Two types of the consensus problem of Julia sets are considered, one is with a leader and the other is with no leaders. Based on these two types, two different control protocols are proposed respectively to make systems achieve consensus of Julia sets. The simulations confirm the efficacy of control protocols.