For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincare-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological "rotational" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.

中文翻译：

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用于平移线和素数端应用的庞加莱-本迪克森定理

对于球体的保向同胚，我们证明如果平移线不在固定点累积，那么它必然会螺旋朝向拓扑吸引子。这与 Poincare-Bendixson 定理对流线的描述类似。然后，我们将此结果应用于无固定点的不变连续体的研究，特别是圆环和简单连接开集的边界。在这些应用中，我们证明如果这样一个开集 $U$ 的素数端旋转数消失，那么要么在边界上有一个不动点，要么 $U$ 的边界包含在一个有限族的盆地中拓扑“旋转”吸引子。这种描述极大地改进了 Cartwright 和 Littlewood 先前的结果，通过从主要端压缩到周围空间。此外，边界附近的动力学与平面图上的非常简单的模型动力学是半共轭的。其他应用涉及不变连续体的可分解性，以及通过圆线上的周期点实现旋转数。