标题： 厚阿诺德舌区 显示英文标题

标题： Thick Arnold tongues

摘要： 我们引入并研究了一个具有物理动机的问题，该问题表现出有趣且或许出乎意料的数学特征。 细胞流是一种二维哈密顿流，其哈密顿量为$H(x, y) = \cos(x) \cos(y)$。 我们研究了由这种流携带的惰性粒子动力学的一个简单模型，考虑粘性阻力以及额外的恒定外力$(b, a)$的影响。 在零惯性粒子的极限情况下，动力学是哈密顿的，具有$H(x, y) = \cos(x) \cos(y) - ax + by$。 对于很小但非零的$a, \ b $，出现了一些“通道”轨迹，这些轨迹蜿蜒至无穷大，相对测度较小，而大多数轨迹仍保持周期性。 相比之下，对于非零惯性（无论多么小），几乎所有粒子轨迹都会漂移到无穷大。 此外，这种漂移的渐近方向不再与强迫的方向一致，而是成为强迫方向$a/b$的类似康托尔函数，并具有一种意外的特性：该函数的平台占据了一个测度为满的集合。 此外，这个集合的补集的豪斯多夫维数为零。 在以两个参数表示的图中（一个参数是强迫方向$a/b$，另一个是拖曳系数），这导致了阿诺德舌区，这些舌区对应于漂移的有理斜率。 然而，与阿诺德的例子不同，所有舌区的并集的补集具有零测度。 这是通过单调圆映射族的旋转数行为来解释的，这些映射族具有平坦区域。 显示英文摘要

摘要： We introduce and study a physically motivated problem that exhibits interesting and perhaps unexpected mathematical features. A cellular flow is a two-dimensional Hamiltonian flow of the Hamiltonian $H(x, y) = \cos(x) \cos(y)$. We study a simple model of the dynamics of an inertial particle carried by such a flow, subject to viscous drag and to an additional constant external force $(b, a)$. In the limiting case of zero inertia particles the dynamics is Hamiltonian with $H(x, y) = \cos(x) \cos(y) - ax + by$. For small but nonzero $a, \ b $ there appear ``channels" of trajectories that wind their way to infinity, of small relative measure, while most trajectories remain periodic. By contrast, for nonzero inertia, no matter how small, almost all particle trajectories drift to infinity. Moreover, the asymptotic direction of this drift no longer coincides with the direction of forcing, and rather becomes Cantor-like function of the forcing direction $a/b$, and with an unexpected feature: the plateaus of this function occupy a set of full measure. Moreover, the complement to this set has zero Hausdorff dimension. In a two-parameter representation (one parameter being the forcing direction $a/b$, the other the drag coefficient), this gives rise to Arnold tongues, the tongues corresponding to rational slopes of drift. However, unlike Arnold's example, the complement to the union of all tongues has zero measure. This is explained by the behavior of rotation number for monotone families of circle maps with flat spots.