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Self-similarity and spectral dynamics

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 Added by Rongwei Yang
 Publication date 2020
  fields
and research's language is English




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For a tuple $A= (A_0, A_1, ldots , A_n)$ of elements in a unital Banach algebra $mathcal{B}$, its textit{projective (joint) spectrum} $p(A)$ is the collection of $zinmathbb{P}^{n}$ such that $A(z)=z_0A_0+z_1 A_1 + ldots z_n A_n$ is not invertible. If the tuple $A$ is associated with the generators of a finitely generated group, then $p(A)$ is simply called the projective spectrum of the group. This paper investigates a connection between self-similar group representations and an induced polynomial map on the projective space that preserves the projective spectrum of the group. The focus is on two groups: the infinite dihedral group $D_infty$ and the Grigorchuk group ${mathcal G}$ of intermediate growth. The main theorem shows that for $D_infty$ the Julia set of the induced rational map $F$ is equal to the union of the projective spectrum with the extended indeterminacy set. Moreover, the limit function of the iteration sequence ${F^{circ n}}$ on the Fatou set is determined explicitly. The result has an application to the group ${mathcal G}$ and gives rise to a conjecture about its associated Julia set.



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179 - J. E. Pascoe 2019
We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the $n^2$ by $n^2$ matrix $sum{overline{X_i}}otimes{X_i}.$ We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely postive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescus theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.
252 - V.I. Bakhtin , A.V. Lebedev 2021
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