Misiurewicz多项式和动力学单位，第I部分
Misiurewicz polynomials and dynamical units, Part I
摘要：
研究了单临界多项式族$f_{d,c}(z)=z^d+c\in \mathbb{C}[z]$的动力学性质。使$f_{d,c}$具有严格前周期后临界轨道的$c$ -值称为Misiurewicz参数，它们是Misiurewicz多项式的根。这些特殊参数的算术性质在算术和复动力学中都有应用。在本文中，我们研究了一些新的这样的性质。特别地，当$d$是素数幂，$c$是Misiurewicz参数时，我们证明了$f_{d,c}$的后临界轨道上的点之间的算术关系.我们还考虑通过评估Misiurewicz多项式在不同的Misiurewicz参数得到的代数整数，我们问这些代数整数时，代数单位。这个问题自然产生于最近由Buff、Epstein和Koch以及第二作者证明的一些结果。我们对这个问题提出了一个合理的答案，并在许多情况下得到了证明。
英文摘要：
We study the dynamics of the unicritical polynomial family $f_{d,c}(z)=z^d+c\in \mathbb{C}[z]$. The $c$-values for which $f_{d,c}$ has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when $d$ is a prime power and $c$ is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of $f_{d,c}$. We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases.
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