We investigate polynomial endomorphisms of graph $C^*$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.

中文翻译：

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图代数的多项式同态

我们研究图$ C ^ * $-代数和Leavitt路径代数的多项式内同态。为此，我们定义和分析与每个这样的内同态对应的编码图。我们找到了一个内在条件仅限于对角MASA自构的条件，当且仅当时，这是根据编码图上某些标签的同步性来表示的。我们表明，这种在无限路径（MASA的频谱）空间上感应出的动力学是由异步换能器产生的。