Sums of two self-similar Cantor sets

Yuki Takahashi

doi:10.1016/j.jmaa.2019.04.051

Sums of two self-similar Cantor sets

Yuki Takahashi

Advanced Institute for Materials Research (AIMR)

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We show that for any pair of self-similar Cantor sets with sum of Hausdorff dimensions greater than 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of self-similar Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture.

Original language	English
Pages (from-to)	613-626
Number of pages	14
Journal	Journal of Mathematical Analysis and Applications
Volume	477
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2019.04.051
Publication status	Published - 2019 Sep 1

Keywords

Cantor sets
Hausdorff dimension
Self-similar sets

ASJC Scopus subject areas

Analysis
Applied Mathematics

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title = "Sums of two self-similar Cantor sets",

abstract = "We show that for any pair of self-similar Cantor sets with sum of Hausdorff dimensions greater than 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of self-similar Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture.",

keywords = "Cantor sets, Hausdorff dimension, Self-similar sets",

author = "Yuki Takahashi",

note = "Funding Information: Y. T. was supported by the Israel Science Foundation grant 396/15 (PI: B. Solomyak). The author is grateful to Boris Solomyak for many helpful discussions and comments.",

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PY - 2019/9/1

Y1 - 2019/9/1

N2 - We show that for any pair of self-similar Cantor sets with sum of Hausdorff dimensions greater than 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of self-similar Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture.

AB - We show that for any pair of self-similar Cantor sets with sum of Hausdorff dimensions greater than 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of self-similar Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture.

KW - Cantor sets

KW - Hausdorff dimension

KW - Self-similar sets

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JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

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ER -

Sums of two self-similar Cantor sets

Abstract

Keywords

ASJC Scopus subject areas

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