Joint spectrum, group representations, and Julia set

2023/01/04 by Yang, Rongwei
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2301.01634

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 π is the Koopman representation of the infinite dihedral group D_∞ on the binary tree, it shows that the projective spectrum of D_∞ coincides with the Julia set of a rational map F_π: ℙ²→ ℙ² derived from the self-similarity of π. This improves the main result in [16].

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