Abstract
In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 - F1 does not contain any intervals but ℒeb(F2 - F1)> 0 almost surely, conditioned on non-extinction.
Original language | English |
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Pages (from-to) | 131-149 |
Number of pages | 19 |
Journal | Indagationes Mathematicae |
Volume | 20 |
Issue number | 1 |
DOIs | |
State | Published - Mar 2009 |
Externally published | Yes |
Bibliographical note
Funding Information:MSC: Primary 28A80; Secondary 60J80, 60J85 Key words and phrases: Random fractals, Difference of Cantor sets, Palis conjecture, Branching processes with random environment ✩P. Móra was supported by OTKA Foundation #TS 49835. K. Simon was supported by OTKA Foundation #K 71693. B. Solomyak was supported by NSF grant DMS-0654408. E-mails: [email protected] (P. Móra), [email protected] (K. Simon), [email protected] (B. Solomyak).
Funding
MSC: Primary 28A80; Secondary 60J80, 60J85 Key words and phrases: Random fractals, Difference of Cantor sets, Palis conjecture, Branching processes with random environment ✩P. Móra was supported by OTKA Foundation #TS 49835. K. Simon was supported by OTKA Foundation #K 71693. B. Solomyak was supported by NSF grant DMS-0654408. E-mails: [email protected] (P. Móra), [email protected] (K. Simon), [email protected] (B. Solomyak).
Funders | Funder number |
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National Science Foundation | DMS-0654408 |
Directorate for Mathematical and Physical Sciences | 0654408 |
Hungarian Scientific Research Fund | 49835, 71693 |
Keywords
- Branching processes with random environment
- Difference of Cantor sets
- Palis conjecture
- Random fractals