自相似性与谱动力学
Self-similarity and spectral dynamics
摘要：
对于单位Banach代数$\mathcal{B}$中元素的元组$A= (A_0, A_1, \ldots , A_n)$，其\textit{projective (joint) spectrum} $p(A)$是$z\in\mathbb{P}^{n}$的集合，使得$A(z)=z_0A_0+z_1 A_1 + \ldots z_n A_n$不可逆。如果元组$A$与一个群的生成元相关联，则$p(A)$简称为群的投影谱。研究了射影空间上自相似群表示与保持群的射影谱的诱导多项式映射之间的关系。重点是两个群体：无限二面角群$D_\infty$和Grigorchuk集团${\mathcal G}$的中间增长。主要定理表明：对于$D_\infty$，诱导有理映射$F$的Julia集等于投影谱与广义不定集的并.此外，还明确地确定了迭代序列$\{F^{\circ n}\}$在Fatou集上的极限函数.结果有一个应用程序的组${\mathcal G}$，并引起一个猜想，其相关的朱莉娅集。
英文摘要：
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 $z\in\mathbb{P}^{n}$ 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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