Spectral dynamics for the Basilica group

The projective spectrum of a tuple $A=(A_0,A_1,\dots,A_n)$ with elements in a unital Banach algebra $\mathcal{A}$ is the collection of $[z_0:z_1:\cdots:z_n]\in \mathbb{P}^n$ such that $z_0A_0+z_1A_1+\dots+z_nA_n$ is not invertible in $\mathcal{A}$. For the tuple $A_ρ=(ρ({\bf e}),ρ({\bf a}),ρ({\bf b}))$, where ${\bf e},{\bf a},{\bf b}$ are the three states of the automaton generating the Basilica group $\mathcal{B}$ and $ρ$ is the Koopman representation, a dynamical map $F$ preserving the projective spectrum of $A_ρ$ is established. Further, a substantial common subset of the Julia set of $F$ and the projective spectrum of $A_ρ$, as well as a large common subset of the Fatou set of $F$ and the projective resolvent set of $A_ρ$ is identified. The result provides partial support for the conjecture of equality of the Julia set of $F$ and the projective spectrum of $A_ρ$.