Hausdorff Dimension of Cantor Intersections and Robust Heterodimensional Cycles for Heterochaos Horseshoe Maps

DOI DOI 被引用文献1件 参考文献27件
  • Hiroki Takahasi
    Keio Institute of Pure and Applied Sciences (KiPAS), Department of Mathematics, Keio University, Yokohama, 223-8522, Japan.
  • James A. Yorke
    Institute for Physical Science and Technology and Mathematics and Physics Departments, University of Maryland, College Park, MD 20742, USA.
  • Yoshitaka Saiki
    Graduate School of Business Administration, Hitotsubashi University, Tokyo, 186-8601, Japan.

書誌事項

公開日
2023-07-28
資源種別
journal article
DOI
  • 10.1137/22m1504986
  • 10.48550/arxiv.2011.12264
公開者
Society for Industrial & Applied Mathematics (SIAM)

説明

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.

25 pages, 9 figures

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