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Topological structure of the sum of two homogeneous Cantor sets

Part of: Classical measure theory Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  24 March 2022

MEHDI POURBARAT
MEHDI POURBARAT*
Affiliation:
Department of Mathematics, Shahid Beheshti University, Tehran, Iran
*
e-mail: m-pourbarat@sbu.ac.ir
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Abstract

We show that in the context of homogeneous Cantor sets, there are generically five possible (open and dense) structures for their arithmetic sum: a Cantor set, an L, R, M-Cantorval and a finite union of closed intervals. The dense case has been dealt with previously. In this paper, we explicitly present pairs of this space which have stable intersection, while not satisfying the generalized thickness test. Also, all the pairs of middle homogeneous Cantor sets whose arithmetic sum is a closed interval are identified.

Keywords

homogeneous Cantor setsPalis conjectureNewhouse’s thickness

MSC classification

Primary: 28A80: Fractals 28A78: Hausdorff and packing measures
Secondary: 37G25: Bifurcations connected with nontransversal intersection
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 5 , May 2023 , pp. 1712 - 1736
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Dedicated to Carlos Gustavo Moreira

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