Universal hierarchical bifurcation and phase transitions in asymmetric Potts models on Cayley trees: exact enumeration for q=5,6,7 and a unified framework for arbitrary q

Abstract

We investigate the hierarchical bifurcation and fixed-point structure of discrete-time dynamical systems arising from the q-state asymmetric Potts model on a Cayley tree of order k=2. For arbitrary q⩾3, the model reduces to a d=q−1-dimensional rational map. We derive a universal quadratic equation governing the fixed points on all synchronized invariant slices, yielding the critical thresholds Kc(d,r)=4r(d−r+1), where K=(θ−1)(ρ−1) with θ=eβJ1 and ρ=eβJ2, and the corresponding double roots u0(r)=(d−r+1)/r. The duality Kc(d,r)=Kc(d,d−r+1) pairs complementary synchronization patterns and reduces the number of distinct critical thresholds to ⌈(q−1)/2⌉. In the generic fully developed regime, K>maxrKc(d,r) with K≠q2, the number of synchronized fixed points reaches 2q−1, yielding the corresponding enumeration of synchronized translation-invariant splitting Gibbs measures under the boundary law correspondence. The exceptional value K=q2 produces a degeneracy in which a quadratic root coincides with the persistent trivial fixed-point u=1. For even q, this value coincides with the critical threshold of the unique self-dual slice. The q=5 model is analyzed in detail, exhibiting successive bifurcations at K=16 and K=24, while the q=6 and q=7 cases further confirm the universal hierarchical structure and its synchronized Gibbs measure enumeration.