Self-similarity and spectral dynamics

This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve spectrum of group. The focus is infinite dihedral D?. main theorem states that Julia set map F P2 for D? union with F's extended indeterminacy set. Moreover, limit function iteration sequence {F?n} Fatou fully described. discovery finds an application to Grigorchuk G intermediate growth its P4. In end, proposes conjecture G's contained in G.