论文标题

Hethochaos马皮地图

Hausdorff dimension of Cantor intersections and robust heterodimensional cycles for heterochaos horseshoe maps

论文作者

Saiki, Yoshitaka, Takahasi, Hiroki, Yorke, James A.

论文摘要

作为在第三维度提供动手动力动力学的基本理解的模型,我们引入了$ c^2 $开放的差异性$ \ Mathbb r^3 $具有两个具有不同不稳定维度的马蹄铁的差异性。我们证明了:一组不稳定的马蹄铁和另一个马stable套件是豪斯多夫尺寸的近2美元,其横截面是Cantor套装;不稳定和稳定组的交集包含一组豪斯多夫尺寸的组合,将近$ 1 $。作为推论,我们检测到$ c^2 $ - 抛光异二维循环。我们的证明采用了通常双曲线不变的歧管和cantor套件的厚度的理论。

As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly $2$ whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. As a corollary we detect $C^2$-robust heterodimensional cycles. Our proof employs the theory of normally hyperbolic invariant manifolds and the thicknesses of Cantor sets.

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