Self-similarity and spectral dynamics

For a tuple $A= (A_0, A_1, \ldots , A_n)$ of elements in a unital Banach algebra $\mathcal{B}$, its \textit{projective (joint) spectrum} $p(A)$ is the collection of $z\in\mathbb{P}^{n}$ such that $A(z)=z_0A_0+z_1 A_1 + \ldots z_n A_n$ is not invertible. If the tuple $A$ is associated with the generators of a finitely generated group, then $p(A)$ is simply called the projective spectrum of the group. This paper investigates a connection between self-similar group representations and an induced polynomial map on the projective space that preserves the projective spectrum of the group. The focus is on two groups: the infinite dihedral group $D_\infty$ and the Grigorchuk group ${\mathcal G}$ of intermediate growth. The main theorem shows that for $D_\infty$ the Julia set of the induced rational map $F$ is equal to the union of the projective spectrum with the extended indeterminacy set. Moreover, the limit function of the iteration sequence $\{F^{\circ n}\}$ on the Fatou set is determined explicitly. The result has an application to the group ${\mathcal G}$ and gives rise to a conjecture about its associated Julia set.