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Nested Cantor sets

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Abstract

We give sufficient conditions for two Cantor sets of the line to be nested for a positive set of translation parameters. This problem occurs in diophantine approximations. It also occurs as a toy model of the parameter selection for non-uniformly hyperbolic attractors of the plane. For natural Cantors sets, we show that this condition is optimal.

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Notes

  1. Points which have a neighborhood whose intersection with \({\hat{K}}\) is uncountable.

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Acknowledgments

We thank the referees for their advices and comments. This research was partially supported by the Balzan project of J. Palis, the French–Brazilian network, the CNPq and the Project BRNUH of Université Sorbonne Paris Cité.

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Authors and Affiliations

  1. LAGA Université Paris 13, Villetaneuse, France

    Pierre Berger

  2. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

    Carlos Gustavo Moreira

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  1. Pierre Berger
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  2. Carlos Gustavo Moreira
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Correspondence to Pierre Berger.

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Berger, P., Moreira, C.G. Nested Cantor sets. Math. Z. 283, 419–435 (2016). https://doi.org/10.1007/s00209-015-1605-6

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  • DOI: https://doi.org/10.1007/s00209-015-1605-6

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