Counting $$\beta $$ -expansions and the absolute continuity of Bernoulli convolutions

Kempton, Tom

doi:10.1007/s00605-013-0512-3

Cited by 17 publications

(28 citation statements)

References 17 publications

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“…Note that given β ∈ (1, 2), almost every x ∈ (0, 1/(β − 1)) has a continuum of β-expansions [12], and furthermore, this continuum can be chosen to have an exponential growth [7].…”

Section: Open Questionsmentioning

confidence: 99%

On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

Hare¹,

Sidorov²

2015

Ergod. Th. Dynam. Sys.

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Let β 1 , β 2 > 1 and T i (x, y) = x+i β1 , y+i β2 , i ∈ {±1}. Let A := A β1,β2 be the unique compact set satisfying A = T 1 (A) ∪ T −1 (A). In this paper we give a detailed analysis of A, and the parameters (β 1 , β 2 ) where A satisfies various topological properties. In particular, we show that if β 1 < β 2 < 1.202, then A has a nonempty interior, thus significantly improving the bound from [2]. In the opposite direction, we prove that the connectedness locus for this family studied in [15] is not simply connected. We prove that the set of points of A which have a unique address has positive Hausdorff dimension for all (β 1 , β 2 ). Finally, we investigate simultaneous (β 1 , β 2 )-expansions of reals, which were the initial motivation for studying this family in [5].

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“…Note that given β ∈ (1, 2), almost every x ∈ (0, 1/(β − 1)) has a continuum of β-expansions [12], and furthermore, this continuum can be chosen to have an exponential growth [7].…”

Section: Open Questionsmentioning

confidence: 99%

On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

Hare¹,

Sidorov²

2015

Ergod. Th. Dynam. Sys.

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“…In [12] we were able to link the growth rate of N n (x; β) for typical x ∈ I β with the question of the absolute continuity of ν β . In particular we defined and f (x) as above but with the lim sup replaced by a lim inf.…”

Section: Equidistribution Resultsmentioning

confidence: 99%

“…To prove this one takes the cover of E β (x) by all cylinders of depth n which intersect E β (x). It was proved in [12] Lemma 3.4, following a similar argument in Appendix C of [18], that lim sup n→∞ β 2 n |{a 1 · · · a n ∈ {0, 1} n : [a 1 · · · a n ] ∩ E β (x) = φ}| ≤ 2h β (x).…”

Section: Hausdorff Measure For Sets Of β-Expansionsmentioning

confidence: 88%

“…We conjecture that the sets O n (x) equidistribute if and only if ν β is absolutely continuous. We are also able to use our results to prove a finer result (Proposition 5.1) about the typical branching rate of sets of β-expansions, making progress towards Conjecture 1 of [12].…”

Section: Introductionmentioning

confidence: 86%

“…Given β ∈ (1, 2), the β-expansion of x ∈ I β is typically not unique, indeed Lebesgue almost every x ∈ I β has uncountably many β-expansions [21]. There is a substantial amount of recent research trying to understand the properties of the sets E β (x) for typical x ∈ I β , see for example [2,3,8,12] and the references therein. Sets of β-expansions can be generated dynamically using the Random β-transformation K β of Dajani and Kraaikamp [6].…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals

Kempton

2016

J. Eur. Math. Soc.

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We study natural measures on sets of β-expansions and on slices through self similar sets. In the setting of β-expansions, these allow us to better understand the measure of maximal entropy for the random β-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.Corresponding theorems due to Marstrand for the dimension of slices through fractals are well known, but the extension to the case of Hausdorff measure of slices through fractals is new.In the final section we state a number of open questions related to our work.

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On universal and periodic β-expansions, and the Hausdorff dimension of the set of all expansions

Baker

2013

Acta Math Hung

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In this paper we study the topology of a set naturally arising from the study of βexpansions. After proving several elementary results for this set we study the case when our base is Pisot. In this case we give necessary and sufficient conditions for this set to be finite. This finiteness property will allow us to generalise a theorem due to Schmidt and will provide the motivation for sufficient conditions under which the growth rate and Hausdorff dimension of the set of β-expansions are equal and explicitly calculable.

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Counting $$\beta $$ -expansions and the absolute continuity of Bernoulli convolutions

Cited by 17 publications

References 17 publications

On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals

On universal and periodic β-expansions, and the Hausdorff dimension of the set of all expansions

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