Let \(\mathbb {F}_q\) be a finite field with q elements and let n be a positive integer. In this paper, we study the digraph associated to the map \(x\mapsto x^n h(x^{\frac{q-1}{m}})\) over \(\mathbb {F}_q\), where \(h(x)\in \mathbb {F}_q[x].\) We completely determine the associated functional graph of maps that satisfy a certain condition of regularity. In particular, we provide the functional graphs associated to monomial maps. As a consequence of our results, one have the number of connected components, length of the cycles and number of fixed points of these class of maps.

中文翻译：

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有限域上多项式映射的动力学

令\(\mathbb {F}_q\)为具有q个元素的有限域，并令n为正整数。在本文中，我们研究了与映射\(x\mapsto x^nh(x^{\frac{q-1}{m}})\)在\(\mathbb {F}_q\)上关联的有向图，其中\(h(x)\in \mathbb {F}_q[x].\)我们完全确定了满足一定规律性条件的映射的关联函数图。特别是，我们提供与单项式映射相关的函数图。根据我们的结果，我们可以得到此类映射的连通分量的数量、循环的长度和固定点的数量。