We provide several new examples in dynamics on the 2‐sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrised families. First, motivated by a topological version of the Poincaré–Bendixson theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of ${\mathbb{S}}^2$ with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arnold's family we construct a parametrised family of Birkhoff‐like cofrontier attractors, where at least for uncountably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This example complements the resolution of Walker's Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a question of Boyland, we show that there exists a non‐transitive Birkhoff‐like attracting cofrontier which is obtained from a Brown–Barge–Martin (BBM) embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor, so that the two induced prime ends rotation numbers are exactly the two end points of the rotation interval. This paper relies extensively on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.

中文翻译：

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素数终结了旋转吸引子参数化族的动力学

我们提供了2球动力学方面的几个新例子，重点是更好地理解参数化族不变域的诱导边界动力学。首先，受Koropecki和Passeggi最近获得的Poincaré–Bendixson定理的拓扑版本的启发，我们证明了存在同胚性。${\mathbb{小号}}^2$和田湖旋转吸引子，具有任意数量的互补域，并且具有或不具有固定点，都与身份任意接近。这回答了勒鲁的问题。其次，从简化的Arnold族中，我们构造了一个参数化的Birkhoff样共边吸引子族，其中至少对于无数的参数选择，两个互补域产生了两个不同的非理性素数旋转数。该示例补充了Koropecki，Le Calvez和Nassiri自2015年以来对Walker猜想的解决方案。第三，回答了Boyland的问题，我们表明存在一个非传递性的Birkhoff样吸引同边，它是从Brown-Barge-Martin获得的（BBM）嵌入圆的逆极限，这样，内部素数终点旋转数就属于共同边界动力学的旋转区间的内部。存在同一个吸引子的另一个BBM嵌入，因此，两个感应的主要端点旋转数正好是旋转间隔的两个端点。本文广泛地依赖于彩色图形。对颜色的某些引用在印刷版本中可能没有意义，我们请读者阅读包含颜色数字的在线版本。