Overlapping Iterated Function Systems from the Perspective of Metric Number Theory
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergenc...
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2023
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| Ausgabe: | 1 |
| Schriftenreihe: | Memoirs of the American Mathematical Society |
| Schlagworte: | |
| ISBN: | 9781470464400, 1470464403 |
| ISSN: | 0065-9266, 1947-6221 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous
result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is
determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated
function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems
that our results apply to include those arising from Bernoulli convolutions, the
For each
Last of all, we introduce a property of an iterated function system that we call being consistently
separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous.
We include several explicit examples of consistently separated iterated function systems. |
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| Bibliographie: | July 2023, volume 287, number 1428 (sixth of 6 numbers) Includes bibliographical references (p. 93-95) |
| ISBN: | 9781470464400 1470464403 |
| ISSN: | 0065-9266 1947-6221 |
| DOI: | 10.1090/memo/1428 |