Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension

Asaoka, Masayuki

doi:10.1090/tran/8452

Stable intersection of Cantor sets in higher dimension and robust homoclinic tangency of the largest codimension
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Trans. Amer. Math. Soc. 375 (2022), 873-908 Request permission

Abstract:

In this paper, we construct

a pair of two regular Cantor sets in the Euclidean space of dimension at least two which exhibits $C^1$-stable intersection, and
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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