We consider reachability decision problems for linear dynamical systems: Given a linear map on $\mathbb{R}^d$ , together with source and target sets, determine whether there is a point in the source set whose orbit, obtained by repeatedly applying the linear map, enters the target set. When the source and target sets are semialgebraic, this problem can be reduced to a point-to-polytope reachability question. The latter is generally believed not to be substantially harder than the well-known Skolem and Positivity Problems. The situation is markedly different for multiple reachability, i.e. the question of whether the orbit visits the target set at least m times, for some given positive integer m. In this paper, we prove that when the source set is semialgebraic and the target set consists of a hyperplane, multiple reachability is undecidable; in fact we already obtain undecidability in ambient dimension d = 10 and with fixed m = 9. Moreover, as we observe that procedures for dimensions 3 up to 9 would imply strong results pertaining to effective solutions of Diophantine equations, we mainly focus on the affine plane ($\mathbb{R}^2$). We obtain two main positive results. We show that multiple reachability is decidable for halfplane targets, and that it is also decidable for general semialgebraic targets, provided the linear map is a rotation. The latter result involves a new method, based on intersections of algebraic subgroups with subvarieties, due to Bombieri and Zannier.

翻译：我们研究线性动力系统的可达性判定问题：给定 $\mathbb{R}^d$ 上的线性映射，以及源集和目标集，确定源集中是否存在一点，使得其通过反复应用该线性映射得到的轨道进入目标集。当源集和目标集为半代数集时，该问题可简化为点-多面体可达性问题。后者通常被认为并不比著名的Skolem问题和Positivity问题更困难。然而，对于多重可达性（即轨道至少访问目标集m次，其中m为给定正整数）的情况，情形则截然不同。在本文中，我们证明：当源集为半代数集且目标集为超平面时，多重可达性是不可判定的；事实上，我们已在环境维度d=10且固定参数m=9的情形下得到不可判定性。此外，由于我们观察到3至9维的判定程序将蕴含关于丢番图方程有效求解的强结果，因此我们主要关注仿射平面($\mathbb{R}^2$)。我们得到两个主要正面结果。我们证明：对于半平面目标，多重可达性是可判定的；且对于一般半代数目标，若线性映射为旋转，则多重可达性也是可判定的。后者涉及一种基于Bombieri-Zannier代数子群与子簇相交理论的新方法。