紧群自同态的跟踪性
Shadowing Endomorphisms of Compact Groups
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中文总结 AI 辅助
该研究刻画了紧Hausdorff群连续自同态的跟踪性，通过约化到紧连通满自同态情形，分别给出了阿贝尔群和一般紧连通群自同态具有跟踪性的等价条件。
中文摘要 AI 辅助
我们刻画了紧Hausdorff群上连续自同态的跟踪性。首先，我们证明了一个自同态具有跟踪性当且仅当其在单位连通分支上的限制具有跟踪性，从而将问题简化为紧连通群的情形。随后通过过渡到稳定像，将分析进一步简化为满自同态的情况。\n 对于紧连通阿贝尔群$A$，设$T_{\mathbb Q}$是由对偶映射在$\widehat A\otimes_{\mathbb Z}\mathbb Q$上诱导的有理线性自同态。我们证明，$A$上的自同态具有跟踪性当且仅当每个有限维$T_{\mathbb Q}$-不变子空间都是双曲的。\n 对于一般的紧连通群，设$A$和$S$分别表示其稳定像的连通中心部分和半单部分。$S/Z(S)$上的诱导自同态确定了单因子集合上的一个单射。我们证明，跟踪性成立当且仅当$A$上的有理对偶映射在每个有限维不变子空间上都是双曲的，且单因子上的诱导映射没有周期点。
英文摘要
We characterize shadowing for continuous endomorphisms of compact Hausdorff groups. First, we prove that an endomorphism has shadowing if and only if its restriction to the identity component does, reducing the problem to compact connected groups. Passing to the stable image then reduces the analysis to surjective endomorphisms. For a compact connected abelian group $A$, let $T_{\mathbb Q}$ be the rational linear endomorphism induced by the dual map on $\widehat A\otimes_{\mathbb Z}\mathbb Q$. We show that the endomorphism of $A$ has shadowing if and only if every finite-dimensional $T_{\mathbb Q}$-invariant subspace is hyperbolic. For a general compact connected group, let $A$ and $S$ denote respectively the connected central and semisimple parts of its stable image. The induced endomorphism on $S/Z(S)$ determines an injective map on the set of simple factors. We prove that shadowing holds exactly when the rational dual map on $A$ is hyperbolic on every finite-dimensional invariant subspace and the induced map on the simple factors has no periodic point.
