Overlapping Iterated Function Systems from the Perspective of Metric Number Theory / Simon Baker.

Format:

Book

Author/Creator:

Baker, Simon, 1846-1919, author.

Series:

Memoirs of the American Mathematical Society ; Volume 287.

Memoirs of the American Mathematical Society ; Volume 287

Language:

English

Subjects (All):

Dynamics--Mathematical models.

Dynamics.

Iterative methods (Mathematics).

Number theory.

Physical Description:

1 online resource (108 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"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 [numbers] problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the [numbers] problem has positive Lebesgue measure. For each [equation] we let [phi]t be the iterated function system given by [equation]. We prove that either [phi]t contains an exact overlap, or we observe Khintchine like behaviour. Our analysis shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures. 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"-- Provided by publisher.

Contents:

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.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Baker, Simon Overlapping Iterated Function Systems from the Perspective of Metric Number Theory

ISBN:

9781470475444

1470475448