论文标题
自相似和光谱动态
Self-similarity and spectral dynamics
论文作者
论文摘要
对于一个元组$ a =（a_0，a_1，\ ldots，a_n）$ $ a _n）$ $ $ a（z）= z_0a_0 + z_1 a_1 + \ ldots z_n a_n $不可逆转。如果元组$ a $与有限生成的组的发电机相关联，则$ p（a）$被简单地称为该组的投影频谱。本文研究了保留该组的投影范围的投影空间上的自相似群体表示与诱导的多项式图之间的联系。重点是两个组：中间增长的无限二面体$ d_ \ infty $和grigorchuk $ {\ Mathcal g} $。主要定理表明，对于$ d_ \ infty $，诱发理性地图$ f $的朱莉娅集合与扩展的不确定性集相同。此外，FATOU集中的迭代序列$ \ {f^{\ circ n} \} $的极限函数是明确确定的。结果对组$ {\ Mathcal G} $进行了应用程序，并引起了有关其相关朱莉亚集合的猜想。
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.
