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

TL;DR

This paper constructs higher-dimensional Cantor sets with stable intersections and demonstrates robust homoclinic tangency of maximal codimension, revealing new phenomena in dynamical systems and answering open questions.

Contribution

It introduces higher-dimensional Cantor sets with stable intersections and constructs hyperbolic sets with robust homoclinic tangency, extending known results to higher dimensions.

Findings

Higher-dimensional Cantor sets can have stable intersections.

Existence of hyperbolic sets with robust homoclinic tangency of largest codimension.

Counterexample to higher-dimensional analogs of Moreira's theorem.

Abstract

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any higher dimensional manifold, using blenders. 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 A.Raibekas.