We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

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关于环面上泛同胚的旋转集

我们研究了圆环上同胚同伦的旋转集$\mathbb T^d$,$d\ge 2$. 在保守设置中，我们证明存在集合的一个 Baire 残差子集$\text {Homeo}_{0, \lambda }(\mathbb T^2)$保守同胚同伦的恒等式，因此具有野生逐点旋转集的点集是Baire残差子集$\mathbb T^2$，并且它具有完整的拓扑压力和完整的度量平均维度。此外，我们证明对于每个$d\ge 2$的旋转集$C^0$-泛型保守同胚$\mathbb T^d$是凸的。在环面上的耗散同胚情况下获得了相关结果。先前的结果依赖于对具有狂野历史行为的点集的拓扑复杂性的描述以及具有粘合轨道特性的连续地图的周期性测量的密集度。