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 $z_0{A}_0+z_1{A}_1+\cdots +z_n{A}_n$ is not invertible in $\mathcal{A}$. For the tuple $A_{\rho }=(\rho (\mathbf{e}),\rho (\mathbf{a}),\rho (\mathbf{b}))$, where $\mathbf{e},\mathbf{a},\mathbf{b}$ are the three states of the automaton generating the Basilica group $\mathcal{B}$ and $\rho$ is the Koopman representation, a dynamical map $F$ preserving the projective spectrum of $A_{\rho }$ is established. Further, a substantial common subset of the Julia set of $F$ and the projective spectrum of $A_{\rho }$, as well as a large common subset of the Fatou set of $F$ and the projective resolvent set of $A_{\rho }$ is identified. The result provides partial support for the conjecture of equality of the Julia set of $F$ and the projective spectrum of $A_{\rho }$.