Spectral dynamics for the infinite dihedral group and the lamplighter group

TL;DR

This work links spectral theory for group representations to complex dynamics by showing that the projective spectrum for an extended pencil associated with the Koopman representation of the infinite dihedral group D_ty coincides with the Julia set of a polynomial map F on 𝔓^3. For the lamplighter group 𝓛, a determinant-recursion approach yields a rational map Q on 𝔓^3 whose Julia set equals the extended indeterminacy set E, and the projective spectrum p(D_π) contains the projection of 𝓛 together with 𝒥(Q). The method hinges on operator recursions from self-similar representations, weak containment/invariance of p(A) under equivalence, and explicit dynamical descriptions via Chebyshev polynomials, enabling precise identifications p(A_π)=𝒥(F) for D_ty and 𝒥(Q)=E for 𝓛. The results illuminate deep connections between spectral dynamics on groups and higher-dimensional complex dynamics, with concrete determinant recursions and semi-conjugacies that facilitate tractable descriptions of spectra and invariant sets.

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 $A(z)=z_0A_0+z_1A_1+\cdots+z_nA_n$ is not invertible. If $\mathfrak{B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_0, A_1,\dots, A_n$ with a representation $ρ$, then $p(A)$ is an invariant of (weak) equivalence for $ρ$. 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 $A_π(z)=z_0+z_1π(a)+z_2π(t)$ with the extended pencil $A_π(z)=z_0+z_1π(a)+z_2π(t)+z_3π(at)$, where $π$ is the Koopman representation, and using the method of operator recursions, we show that $p(A_π)=\mathcal{J}(F).$ 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 $\mathcal{L}$.

Figures (2)

• Figure 1: 2-regular rooted tree
• Figure 2: Automaton of the group $D_\infty$

Theorems & Definitions (31)

• Definition 1.1
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Definition 2.5
• Lemma 3.1: GZ, Lemma 7
• proof
• Lemma 3.2
• proof