Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T18:29:58.254Z Has data issue: false hasContentIssue false
  • English
  • Français

N-fold sums of Cantor sets

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Kathryn E. Hare  and
Toby C. O'Neil
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1. E-mail: [email protected]
Toby C. O'Neil
Affiliation:
Department of Pure Mathematics, Faculty of Maths and Computing, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK. E-mail: [email protected]
Get access

Abstract

The Newhouse gap lemma is generalized by finding a geometric condition which ensures that N-fold sums of compact sets, which might even have thickness zero, are intervals. A new proof is also obtained of a lower bound on the thickness of the sum of two Cantor sets.

MSC classification

Secondary: 28A80: Fractals
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 243 - 250
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Astels, S.. Cantor sets and numbers with restricted partial quotients. Trans. Anier. Math. Soc., 352 (2000), 133–170.CrossRefGoogle Scholar
2.Cabrelli, C., Hare, K. and Molter, U.. Sums of Cantor sets. Ergodic Theory and Dynamical Systems, 17 (1997), 1299–1313.CrossRefGoogle Scholar
3.Moreira, C. and Yoccoz, J.. Stable intersections of Cantor sets with large Hausdorff dimension. Ann. of Math. (2) 154 (2001), 45–96.CrossRefGoogle Scholar
4.Newhouse, S.. Lectures on dynamical systems. In Dynamical Systems, CIME Lecture, Bressanone, Italy, 1978; Progress in Mathematics, 8 (Birkhauser, 1980), 1–114.Google Scholar
5.Palis, J. and Takens, F.. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Cambridge Studies in Adv. Math., 35 (1993).Google Scholar
6.Palis, J. and Yoccoz, J.. On the arithmetic sum of regular Cantor sets. Ann. Inst. H. Poincaré Anal. Nonlineaire, 14 (1997), 439–456.CrossRefGoogle Scholar