Minimal extensions in smooth dynamics

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Abstract

A classical result of Fathi and Herman from 1977 states that a smooth compact connected manifold without boundary admitting a locally free action of a 1-torus, respectively, an almost free action of a 2-torus, admits a minimal diffeomorphism, respectively, a minimal flow. In the first part of our paper we study the existence of locally free and almost free actions of tori on homogeneous spaces of compact connected Lie groups, thus providing new examples of spaces admitting minimal diffeomorphisms or flows. In the second part we combine the ideas of Fathi and Herman with our recent ideas to study the existence of minimal skew products over certain minimal flows with general connected Lie groups as acting groups. Our results apply to so called flows with free cycles. In the last part of our work we study the existence of free cycles in homogeneous flows.

平稳动力学中的最小扩展
摘要 Fathi 和 Herman 1977 年的一个经典结果指出,一个光滑的无边界紧凑连通流形,如果接纳一个 1 次旋的局部自由作用,或一个 2 次旋的几乎自由作用,就会接纳一个最小的衍射,或一个最小的流。在论文的第一部分,我们研究了在紧凑连通李群的同质空间上存在的局部自由和几乎自由的环作用,从而提供了容许极小差分或极小流的空间的新例子。在第二部分中,我们将法蒂和赫尔曼的观点与我们最近的观点相结合,研究了以一般连通李群为作用群的某些极小流上的极小斜积的存在性。我们的结果适用于所谓的自由循环流。在工作的最后一部分,我们研究了同质流中自由循环的存在性。
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