Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension
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Abstract:
In this paper, we construct
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a pair of two regular Cantor sets in the Euclidean space of dimension at least two which exhibits $C^1$-stable intersection, and
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a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any manifold of dimension at least four.
The former implies that an analog of Moreira’s theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and Raibekas.
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Additional Information
- Masayuki Asaoka
- Affiliation: Faculty of Science and Engineering, Doshisha University, Kyotanabe 610-0394, Japan
- MR Author ID: 629522
- Email: masaoka@mail.doshisha.ac.jp
- Received by editor(s): November 25, 2019
- Received by editor(s) in revised form: September 29, 2020, March 29, 2021, and March 30, 2021
- Published electronically: December 3, 2021
- Additional Notes: This paper was partially supported by the JSPS Kakenki Grants 18K03276
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 873-908
- MSC (2020): Primary 37C29; Secondary 28A80
- DOI: https://doi.org/10.1090/tran/8452
- MathSciNet review: 4369238