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 the projective spectrum of the group. The focus is on the infinite dihedral group D_∞. The main theorem states that the Julia set of the induced rational map F on P² for D_∞ is the union of the projective spectrum with F’s extended indeterminacy set. Moreover, the limit function of the iteration sequence (Formula Presented) on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G’s projective spectrum is contained in the Julia set of G.