Singularity versus exact overlaps for self-similar measures

Simon, Károly; Vágó, Lajos

doi:10.1090/proc/14042

“…The claim dim P lim sup E i = d also follows by observing that lim sup E i is a dense G δ -set [SV,Fact 12], but we give a direct proof. We will actually prove the following result; it is clear that Theorem 3.1 is an immediate corollary.…”

Section: Statement Of Resultsmentioning

confidence: 84%

Mass transference principle: from balls to arbitrary shapes

Koivusalo¹,

Rams²

2018

Preprint

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“…A standard technique for proving an attractor has positive d-dimensional Lebesgue measure is by showing there is an absolutely continuous pushforward measure. Note that by a recent result of Simon and Vágó [57], it follows that the list of mechanisms leading to the failure of absolute continuity is strictly greater than the list of mechanisms leading to the failure of equality in (1.2). The usual methods for gauging how an iterated function system overlaps are to determine whether the Hausdorff dimension of the attractor satisfies a certain formula, to determine whether the dimension of pushforwards of dynamically-defined measures satisfy a certain formula, and to determine whether these measures are absolutely continuous with respect to the Lebesgue measure.…”

Section: Attractors Generated By Iterated Function Systemsmentioning

confidence: 97%

Overlapping iterated function systems from the perspective of Metric Number Theory

Baker¹

2019

Preprint

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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. This result 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 our results apply to include those arising from Bernoulli convolutions, the {0, 1, 3} problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the {0, 1, 3} problem has positive Lebesgue measure.For each t ∈ [0, 1] we let Φ t be the iterated function system given by

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Overlapping Iterated Function Systems from the Perspective of Metric Number Theory

Baker¹

2023

Memoirs of the AMS

2

0

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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 { 0 , 1 , 3 } \{0,1,3\} 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 { 0 , 1 , 3 } \{0,1,3\} problem has positive Lebesgue measure. For each t ∈ [ 0 , 1 ] t\in [0,1] we let Φ t \Phi _t be the iterated function system given by Φ t ≔ { ϕ 1 ( x ) = x 2 , ϕ 2 ( x ) = x + 1 2 , ϕ 3 ( x ) = x + t 2 , ϕ 4 ( x ) = x + 1 + t 2 } . \begin{equation*} \Phi _{t}≔\Big \{\phi _1(x)=\frac {x}{2},\phi _2(x)=\frac {x+1}{2},\phi _3(x)=\frac {x+t}{2},\phi _{4}(x)=\frac {x+1+t}{2}\Big \}. \end{equation*} We prove that either Φ t \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.

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Singularity versus exact overlaps for self-similar measures

Cited by 4 publications

References 10 publications

Mass transference principle: from balls to arbitrary shapes

Mass transference principle: from balls to arbitrary shapes

Overlapping iterated function systems from the perspective of Metric Number Theory

Overlapping Iterated Function Systems from the Perspective of Metric Number Theory

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