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010 a| 2023031258;

020 a| 9781470464400q| (paperback);

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100 1 a| Baker, Simonc| (Lecturer in mathematics)?| UNAUTHORIZED;

245 10 a| Overlapping iterated function systems from the perspective of metric number theoryh| [electronic resource] /c| Simon Baker.;

260 a| Providence, RI :b| American Mathematical Society,c| 2023.;

300 a| v, 95 pages :b| illustrations ;c| 26 cm.;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| volume 287, number 1428;

504 a| Includes bibliographical references (pages 93-95).;

505 0 a| 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.;

506 a| Available only to authorized users.;

520 a| "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"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Iterative methods (Mathematics)=| ^A19810;

650 0 a| Number theory.=| ^A22877;

650 0 a| Measure theory.=| ^A21743;

650 0 a| Diophantine approximation.=| ^A213381;

650 0 a| Fractals.=| ^A187351;

650 0 a| Dynamicsx| Mathematical models.=| ^A991442;

650 7 a| Number theory -- Probabilistic theory: distribution modulo $1$; metric theory of algorithms -- Diophantine approximation.2| msc;

650 7 a| Measure and integration -- Classical measure theory -- Fractals.2| msc;

650 7 a| Dynamical systems and ergodic theory -- Smooth dynamical systems: general theory -- Dimension theory of dynamical systems.2| msc;

655 0 a| Electronic books.=| ^A491897;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=30671911;

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Overlapping iterated function systems from the perspective of metric number theory / Simon Baker.

Author/creator	Baker, Simon
Format	Electronic
Publication Info	Providence, RI : American Mathematical Society, 2023.
Description	v, 95 pages : illustrations ; 26 cm.
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Iterative methods (Mathematics) Number theory. Measure theory. Diophantine approximation. Fractals. Dynamics--Mathematical models.

Series	Memoirs of the American Mathematical Society, 0065-9266 ; volume 287, number 1428
Contents	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.
Abstract	"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.
Bibliography note	Includes bibliographical references (pages 93-95).
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2023031258
ISBN	9781470464400 (paperback)
ISBN	(pdf)

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