论文标题
A $ C^r $ - 连接Lorenz吸引者的引理及其在Ergodic措施空间中的应用
A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures
论文作者
论文摘要
对于\ Mathbb {n} _ {\ geq 2} \ cup \ {\ infty \} $的每一个$ r \，我们证明了$ c^r $ - 连接的lorenz吸引者。确切地说，对于洛伦兹的吸引子，具有$ 3 $维的$ c^r $（$ r \ geq 2 $）矢量场，可以通过任意的$ c^r $ perturtations创建与奇异性和关键元素相关的异质轨道。作为一种应用，我们表明，对于$ c^r $浓度的几何洛伦兹吸引子，奇异性的狄拉克度量是隔离的，在ergodic措施的空间内隔离，因此无法连接麦芽糖的测量空间。尽管对于$ c^r $ - 生成的几何洛伦兹吸引子，但千古措施的空间与密集的周期性措施相连。特别是，通用部分证明了Bonatti C. Bonatti在$ C^r $ - 洛伦兹吸引者中提出的猜想。
For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures and thus the ergodic measure space is not connected; while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti in $C^r$-topology for Lorenz attractors.
