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arXiv · 2301.01634

Joint spectrum, group representations, and Julia set

Rongwei Yang
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Abstract

The first half of this mostly expository note reviews some notions of joint spectrum of linear operators, and it gives a new characterization of amenable groups in terms of projective spectrum. The second half revisits an application of projective spectrum to the study of self-similar group representations made in [16]. In the case $\pi$ is the Koopman representation of the infinite dihedral group $D_\infty$ on the binary tree, it shows that the projective spectrum of $D_\infty$ coincides with the Julia set of a rational map $F_\pi: \mathbb{P}^2\to \mathbb{P}^2$ derived from the self-similarity of $\pi$. This improves the main result in [16].

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Rongwei Yang. 2023-01-04. Joint spectrum, group representations, and Julia set. https://arxiv.org/abs/2301.01634

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