Dynamics of rational symplectic mappings and difference Galois theory

A note about rational representations of differential Galois groups

We give a description of the rational representations of the differential Galois group of a Picard-Vessiot extension.

具有非实 Julia 集的有理映射的树状 Galois 群

Arboreal Galois groups of rational maps with nonreal Julia sets

本文研究具有非实 Julia 集的有理映射的树状 Galois 群，证明相关映射及与椭圆曲线倍乘映射关联的 Lattès 映射具有非交换树状 Galois 群，并探讨实系数多项式何时具有实 Julia 集。

A Galois-dynamics correspondence for unicritical polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton-Patel, and Vivaldi-Hatjispyros in the early 1990s con…

具有碰撞临界点的二次有理函数的树状 Galois 群

Arboreal Galois groups for quadratic rational functions with colliding critical points

本文研究了二次有理函数临界点碰撞时的树状 Galois 群，给出了子群 $M_{\ell}$ 的新描述与 Pink 定理的新证明，并证明了 $G_{\infty}$ 成为全群的充要条件。

Non-integrability of some Hamiltonians with rational potentials

In this work we compute the families of classical Hamiltonians in two degrees of freedom in which the Normal Variational Equation around an invariant plane falls in Schroedinger type with polynomial or trigonometrical potential. We analyze the integrability of Normal Variational Equation in Liouvillian sense using the Kovacic's algorithm. We compute all Galois groups of Schroedinger type equations with polynomial potential. We also introduce a method of algebrization that transforms equations with transcendental coefficients in equations with rational coefficients without changing essentially …