非可逆随机动力系统的原子物理测度
Atomic physical measures for non-invertible random dynamical systems
AI总结：
研究非可逆随机动力系统，构造圆上由局部可逆映射构成的系统并证明其有原子物理平稳测度$\nu$，指出平稳测度赫尔德正则性不能推广，还给出相关示例。
AI中文摘要：
我们构造了一个圆上的随机动力系统示例，该系统由仅局部可逆的映射构成，且具有原子平稳测度$\nu$。此外，此测度是物理的：对于几乎所有勒贝格初始点$x_0$，其随机轨迹的切萨罗平均值几乎必然收敛于$\nu$。这表明由微分同胚构成的（非保测）随机动力系统中已知的平稳测度的赫尔德正则性不能推广到这类系统。我们还提供了一些相关示例，包括平稳测度对一个真子流形赋非零测度但不存在封闭公共不变子流形的情况。
英文摘要：
We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure $ν$. Moreover, this measure is physical: for Lebesgue-almost every initial point $x_0$, the Cesàro averages of its random trajectory almost surely converge to $ν$. This shows that the Hölder regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.
