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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| Hlavní autor: | |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Providence, Rhode Island
American Mathematical Society
2023
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| Vydání: | 1 |
| Edice: | Memoirs of the American Mathematical Society |
| Témata: | |
| ISBN: | 9781470464400, 1470464403 |
| ISSN: | 0065-9266, 1947-6221 |
| On-line přístup: | Získat plný text |
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- Introduction -- Statement of results -- Preliminary results -- Applications of Proposition 3.1 -- A specific family of IFSs -- Proof of Theorem 2.15 -- Proof of Theorem 2.16 -- Applications of the mass transference principle -- Examples -- Final discussion and open problems -- Acknowledgments
- Cover -- Title page -- Chapter 1. Introduction -- 1.1. Attractors generated by iterated function systems -- 1.2. Diophantine approximation and metric number theory -- 1.3. Two families of limsup sets -- 1.3.1. The set _{Φ}( ,Ψ) -- 1.3.2. The set _{Φ}( ,\m,ℎ) -- Chapter 2. Statement of results -- 2.1. Parameterised families with variable contraction ratios -- 2.2. Parameterised families with variable translations -- 2.3. A specific family of IFSs -- 2.3.1. New methods for distinguishing between the overlapping behaviour of IFSs -- 2.4. The CS property and absolute continuity. -- 2.5. Overlapping self-conformal sets -- 2.6. Structure of the paper -- Chapter 3. Preliminary results -- 3.1. A general framework -- 3.1.1. Verifying the hypothesis of Proposition 3.1. -- 3.1.2. The non-existence of a Khintchine like result -- 3.2. Full measure statements -- Chapter 4. Applications of Proposition 3.1 -- 4.1. Proof of Theorem 2.2 -- 4.1.1. Bernoulli convolutions -- 4.1.2. The {0,1,3} problem -- 4.2. Proof of Theorem 2.9 -- Chapter 5. A specific family of IFSs -- Chapter 6. Proof of Theorem 2.15 -- Chapter 7. Proof of Theorem 2.16 -- Chapter 8. Applications of the mass transference principle -- Chapter 9. Examples -- 9.1. IFSs satisfying the CS property -- 9.2. The non-existence of Khintchine like behaviour without exact overlaps -- Chapter 10. Final discussion and open problems -- Acknowledgments -- Bibliography -- Back Cover