Spectral dynamics for the infinite dihedral group and the lamplighter group

Abstract

For a tuple $A=(A_0,A_1,\cdots ,A_n)$ of elements in a Banach algebra $\mathfrak{B}$, its projective (joint) spectrum $p(A)$ is the collection of $z\in {\mathbb{P}}^n$ such that $A(z)=z_0{A}_0+z_1{A}_1+\cdots +z_n{A}_n$ is not invertible. If $\mathfrak{B}$ is the group $C^{\ast }$-algebra for a discrete group $G$ generated by $A_0,A_1,\dots ,A_n$ with a representation $\rho$, then $p(A)$ is an invariant of (weak) equivalence for $\rho$. In \cite{BY}, B. Goldberg and R. Yang proved that the Julia set $\mathcal{J}(F)$ of the induced rational map $F$ for the infinite dihedral group $D_{\infty }$ is the union of the projective spectrum with the extended indeterminacy set. But the extended indeterminacy set $E_F$ is complicated. To obtain a better relationship between the projective spectrum and the Julia set, by replacing $A_{\pi }(z)=z_0+z_1\pi (a)+z_2\pi (t)$ with the extended pencil $A_{\pi }(z)=z_0+z_1\pi (a)+z_2\pi (t)+z_3\pi (at)$, where $\pi$ is the Koopman representation, and using the method of operator recursions, we show that $p(A_{\pi })=\mathcal{J}(F).$ Further, we study the spectral dynamics for the Lamplighter group $\mathcal{L}$, and prove that $\mathcal{J}(Q)=E_Q$, where $Q$ is the rational map associated with $\mathcal{L}$.

arXiv comment20 pages, 2 figures