论文标题
MISIUREWICZ多项式和动力单位，第一部分
Misiurewicz polynomials and dynamical units, Part I
论文作者
论文摘要
我们研究单一政治多项式家族$ f_ {d，c}（z）= z^d+c \ in \ mathbb {c} [z] $的动力学。 $ f_ {d，c} $的$ c $ - 价值严格的前临界轨道称为Misiurewicz参数，它们是Misiurewicz多项式的根源。这些特殊参数的算术特性在算术和复杂动力学中都发现了应用。在本文中，我们研究了一些新的此类属性。特别是，当$ d $是主要功率，而$ c $是误导性参数时，我们证明了$ f_ {d，c} $的临界轨道中的点之间的某些算术关系。我们还考虑通过在不同的Misiurewicz参数上评估误导性多项式获得的代数整数，我们询问这些代数整数何时为代数单位。这个问题自然来自Buff，Epstein和Koch最近证明的一些结果以及第二作者。我们对这个问题提出了一个猜想的答案，在许多情况下，我们证明了这一点。
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.
