AI 中文总结
本文针对生成Basilica群的自动机三状态对应的Koopman表示的n元组，建立保持其射影谱的动力映射，识别出该映射的Julia集、Fatou集与对应射影谱、射影预解集的大量公共子集，为相关相等猜想提供部分支撑。
AI 中文摘要
带有元素的n元组$A=(A_0,A_1,\dots,A_n)$在单位巴拿赫代数$\mathcal{A}$中的射影谱是满足$z_0A_0+z_1A_1+\dots+z_nA_n$在$\mathcal{A}$中不可逆的$[z_0:z_1:\cdots:z_n]\in \mathbb{P}^n$的集合。对于n元组$A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$，其中${\bf e},{\bf a},{\bf b}$是生成Basilica群$\mathcal{B}$的自动机的三个状态，$\rho$是Koopman表示，本文建立了保持$A_{\rho}$射影谱的动力映射$F$。进一步，识别出$F$的Julia集与$A_{\rho}$射影谱的一个大量公共子集，以及$F$的Fatou集与$A_{\rho}$射影预解集的一个大量公共子集。该结果为$F$的Julia集与$A_{\rho}$射影谱相等的猜想提供了部分支撑。
英文摘要
The projective spectrum of a tuple $A=(A_0,A_1,\dots,A_n)$ with elements in a unital Banach algebra $\mathcal{A}$ is the collection of $[z_0:z_1:\cdots:z_n]\in \mathbb{P}^n$ such that $z_0A_0+z_1A_1+\dots+z_nA_n$ is not invertible in $\mathcal{A}$. For the tuple $A_ρ=(ρ({\bf e}),ρ({\bf a}),ρ({\bf b}))$, where ${\bf e},{\bf a},{\bf b}$ are the three states of the automaton generating the Basilica group $\mathcal{B}$ and $ρ$ is the Koopman representation, a dynamical map $F$ preserving the projective spectrum of $A_ρ$ is established. Further, a substantial common subset of the Julia set of $F$ and the projective spectrum of $A_ρ$, as well as a large common subset of the Fatou set of $F$ and the projective resolvent set of $A_ρ$ is identified. The result provides partial support for the conjecture of equality of the Julia set of $F$ and the projective spectrum of $A_ρ$.
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