We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

中文翻译：

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除环上多项式的定点

我们研究标准（或左）多项式的离散动力学$f(x)$在分割环上D. 我们将它们的固定点定义为点$\lambda \in D$为此$f^{\circ n}(\lambda )=\lambda $对于任何$n \in \mathbb {N}$， 在哪里$f^{\circ n}(x)$递归定义为$f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$和$f^{\circ 1}(x)=f(x)$. 周期点的定义类似。我们证明$\λ$是一个不动点$f(x)$当且仅当$f(\lambda )=\lambda $，这使得能够使用多项式方程理论的已知结果来得出结论，任何次数的多项式$m \geq 2$最多有米不动点的共轭类。我们还表明，一般情况下，周期点的行为与交换情况不同。我们为周期点按预期表现提供了充分条件。