There are no C1-stable intersections of regular Cantor sets

Abstract

We prove that there are no stable intersections of regular Cantor sets in the C¹ topology: given any pair (K, K′) of regular Cantor sets, we can find, arbitrarily close to it in the C¹ topology, pairs $ \left( {\tilde{K},\tilde{K}'} \right) $ of regular Cantor sets with $ \tilde{K} \cap \tilde{K}' = \emptyset $.

Moreover, for generic pairs (K, K′) of C¹-regular Cantor sets, the arithmetic difference $ K - K' = \left\{ {x - y:x \in K,y \in K'} \right\} = \left\{ {t \in \mathbb{R}:K \cap \left( {K' + t} \right) \ne \emptyset } \right\} $ has empty interior (and so is a Cantor set). This is very different from the situation in the C² topology: according to a theorem by Moreira and Yoccoz, typical pairs (K, K′) of C²-regular Cantor sets whose sum of Hausdorff dimensions is larger than 1 are such that their arithmetic difference K, K′ is the closure of its interior.

We also show that there is a generic set $ \mathcal{R} $ of C¹ diffeomorphisms of M such that, for every $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\psi }} \in \mathcal{R} $, there are no tangencies between leaves of the stable and unstable foliations of Λ₁, Λ₂, for any horseshoes Λ₁, Λ₂ of $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\psi }} $.