The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression ratio. For example, it turns out that the spectral radius of the operator

is equal to \(2p^{n/2}\) for transcendental values of \(p\), and depends on the coefficients of the minimal polynomial for \(p\) in the case where \(p\) is an algebraic number. In this paper, we study this dependence. The starting point is the well-known statement that, given a velocity vector with rationally independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for \(p\). The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of \(p\).

具有移位和压缩参数的泛函微分方程的边值问题唯一可解性的条件通过对应类泛函算子的谱半径公式表示。该公式的使用涉及某些类型限制的计算，即使在最简单的情况下，这些限制也表现出对压缩比的惊人“混乱”依赖性。例如，结果证明算子的谱半径

等于\(2p^{n/2}\)对于\(p\) 的超越值，并且在\(p\)是\(p\)的情况下取决于\(p\)的最小多项式的系数代数数。在本文中，我们研究了这种依赖性。出发点是众所周知的陈述，给定一个具有合理独立坐标的速度矢量，相应的线性流在环面上是最小的，即每个点的轨迹在环面上到处都是密集的。我们证明了这个陈述的一个版本，它有助于在合理依赖速度的情况下控制轨迹的行为。对于最小多项式的系数的各种情况，获得谱半径的上限和下限\(p\)。该论文的主要结果是\(p\) 的有理（和任何有理数的根）值的谱半径的精确公式。