稳定的局部动力学：扩张、拟共形性与遍历性
Stable local dynamics: expansion, quasi-conformality and ergodicity
AI总结：
本文引入拟共形混合器这一几何方法，克服了稳定遍历作用仅存在于一维流形的限制，证明了闭流形上$C^{1+α}$类微分同胚的有限生成群作用及某些代数作用的稳定遍历性。
摘要：
本文研究微分同胚群在光滑流形上作用的稳定遍历性。已知此类作用仅存在于一维流形。本文旨在引入一种几何方法以克服这一限制，并构造高维例子。特别地，我们证明每个闭流形都允许通过$C^{1+α}$类微分同胚进行稳定遍历的有限生成群作用。我们还证明了某些代数作用的稳定遍历性，包括在任意维球面上靠近单位元的矩阵对的自然作用。这些结果是拟共形混合器的推论，这是本文引入的一种局部且稳定的机制/现象，它通过提供具有精细控制几何的拟共形轨道来概括我们证明稳定局部遍历性的方法。拟共形混合器是在局部定义的微分同胚的伪半群作用背景下发展起来的，允许在多种环境中应用。
英文摘要：
In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class $C^{1+α}$. We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the quasi-conformal blender, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.
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