迭代伽罗瓦群的逆伽罗瓦问题及其不动点比例
The inverse Galois problem of iterated Galois groups and their fixed-point proportion
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中文总结 AI 辅助
本文引入几乎混合群概念，研究迭代伽罗瓦群的逆伽罗瓦问题，证明非动力学拉回的有理函数几何迭代伽罗瓦群不动点比例为零，解决相关开放问题。
中文摘要 AI 辅助
1985年，Odoni受算术动力学中素数密度问题的驱动，开启了对树状表示与不动点比例的研究。此后，人们提出了诸多关于有理函数动力学与其动力学对应的伽罗瓦群（迭代伽罗瓦群）之间关联的问题。本文对这一关联作出了若干贡献，引入了几乎混合群的概念。这一新概念与迭代伽罗瓦群的自复制能力相关，当迭代伽罗瓦群无法完全自复制时，其失效原因总是一个有限指数正规子群。该子群的商群包含了有理函数性质的重要信息。首先，我们研究逆伽罗瓦问题，证明有理函数的迭代伽罗瓦群总是几乎混合的，而失效情况（即该有限指数子群的商群）与有理函数的几何性质相关。作为推论，我们证明：当且仅当该失效情况非平凡时，有理函数由代数曲线的自同态（动力学拉回）诱导。最后，我们解决了有理函数几何迭代伽罗瓦群不动点比例的主要开放问题，证明当映射不是动力学拉回时，不动点比例为零。该结果可直接应用于素数密度问题、映射约简中的周期点比例以及有限域上的周期点比例。本文的证明结合了群论、遍历理论、概率论、数论、复动力学、图的能量传输、算术动力学与代数几何的技术和结果。
英文摘要
In 1985, Odoni initiated the study of arboreal representations and the fixed-point proportion, motivated by prime density problems in arithmetic dynamics. Since then, many questions regarding the connection between the dynamics of rational functions and the Galois groups associated to their dynamics (iterated Galois groups) have been posed. In this article, we give several contributions to this connection with the introduction of the concept of virtually mixing groups. This new concept is related to the capability of self-replication of the iterated Galois groups, and when it fails to be completely self-replicated, the failure is always by a finite index normal subgroup. It turns out that the quotient by this subgroup contains valuable information of the properties of the rational function. First, we investigate the inverse Galois problem, showing that iterated Galois groups of rational functions are always virtually mixing and the failure (the quotient by this finite-index subgroup) is related to geometric properties of the rational function. As a consequence, we prove that the rational function is induced by an endomorphism of an algebraic curve (dynamical pullback) if and only if this failure is non-trivial. Finally, we solve the main open problem of the fixed-point proportion of geometric iterated Galois groups of rational functions, by showing that the fixed-point proportion is zero when the map is not a dynamical pullback. This result has direct applications to prime density problems, proportion of periodic points in the reduction of maps and proportion of periodic points over finite fields. The proofs in this article rely on a combination of techniques and results from group theory, ergodic theory, probability, number theory, complex dynamics, energy transport in graphs, arithmetic dynamics and algebraic geometry.
