dorsal/arxiv
View SchemaSpectral dynamics for the infinite dihedral group and the lamplighter group
| Authors | Chao Zu, Yixin Yang, Yufeng Lu |
|---|---|
| Categories | |
| ArXiv ID | 2601.09155vv1 |
| URL | https://arxiv.org/abs/2601.09155 |
| Journal | Indiana University Mathematics Journal, VOLUME = {73}, YEAR = {2024}, NUMBER = {3} |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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 $\rho$, 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 $\mathcal{L}$.
{
"annotation_id": "737de408-df9e-492b-99b9-999a4e85a5a6",
"date_created": "2026-02-17T05:53:19.854000Z",
"date_modified": "2026-02-17T05:53:19.854000Z",
"file_hash": "099bab2d31392bca5bf58344b2dd832e6d4e4c0c2a5f9ab2053927ff7520e25b",
"private": false,
"record": {
"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 $\\rho$, 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_id": "2601.09155",
"authors": [
"Chao Zu",
"Yixin Yang",
"Yufeng Lu"
],
"categories": [
"math.FA",
"math.DS",
"math.SP"
],
"journal_ref": "Indiana University Mathematics Journal, VOLUME = {73}, YEAR = {2024}, NUMBER = {3}",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Spectral dynamics for the infinite dihedral group and the lamplighter group",
"url": "https://arxiv.org/abs/2601.09155",
"version": "v1"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c320f42d-5d7c-4797-9ddb-1aea31dbfb06",
"id": "arXiv Dataset",
"type": "Model",
"variant": "snapshot-2026-01-17",
"version": "0.1.0"
},
"user_id": 1000002
}