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2011 There are no C1-stable intersections of regular Cantor sets
Carlos Gustavo Moreira
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Acta Math. 206(2): 311-323 (2011). DOI: 10.1007/s11511-011-0064-0

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 }} $.

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Carlos Gustavo Moreira. "There are no C1-stable intersections of regular Cantor sets." Acta Math. 206 (2) 311 - 323, 2011. https://doi.org/10.1007/s11511-011-0064-0

Information

Received: 7 April 2009; Revised: 30 April 2010; Published: 2011
First available in Project Euclid: 31 January 2017

zbMATH: 1251.37032
MathSciNet: MR2810854
Digital Object Identifier: 10.1007/s11511-011-0064-0

Rights: 2011 © Institut Mittag-Leffler

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Vol.206 • No. 2 • 2011
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