There are no C1-stable intersections of regular Cantor sets

By Carlos Gustavo T. de A. Moreira. 2009

53 citations recorded by the source corpus.

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 }} $.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/55717acb21d49efe9a0ca3ae48e6c5e0499f36ee

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In Geometry and Topology

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Citations in the corpus

Abundance of C 1 -robust homoclinic tangencies (2009, background): “As mentions above, Moreira’s result in [17] provides strong evidences suggesting that O1(M 2) is empty.”
Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures (2015, background): “f C1-robust homoclinic tangencies occur in dimension at least 3 and are associated to C1-robust cycles: see [BD3]. Furthermore, C1-surface dieomorphisms do not have robust homoclinic tangencies: see [Mo]. 4 J. BOCHI, C. BONATTI, AND L.J. DIAZ contains an open and dense subset of the set of C1-robust”