# Journal of Operator Theory

Volume 87, Issue 2, Spring 2022 pp. 355-388.

Self-similarity and spectral dynamics

**Authors**:
Bryan Goldberg (1), Rongwei Yang (2)

**Author institution:** (1) Department of Mathematics, State University of New York At Albany, Albany, NY 12222, U.S.A.

(2) Department of Mathematics, State University of New York At Albany, Albany, NY 12222, U.S.A.

**Summary: **This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group $D_\infty$. The main theorem states that the Julia set of the induced rational map $F$ on ${\mathbb P}^2$ for $D_\infty$ is the union of the projective spectrum with $F$'s extended indeterminacy set. Moreover, the limit function of the iteration sequence $\{F^{\circ n}\}$ on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group ${\mathcal G}$ of intermediate growth and its induced rational map $G$ on ${\mathbb P}^4$. In the end, the paper proposes the conjecture that ${\mathcal G}$'s projective spectrum is contained in the Julia set of $G$.

**DOI: **http://dx.doi.org/10.7900/jot.2020sep27.2329

**Keywords: **projective spectrum, self-similar representation, infinite dihedral group, Grigorchuk group, indeterminacy set, Fatou set, Julia set, Tchebyshev polynomial

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