Recent progress on Bernoulli convolutions

Varjú, Péter P.

doi:10.4171/176-1/38

Cited by 27 publications

(28 citation statements)

References 55 publications

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“…with σ i = 0 and σ i σ j = δ i,j . From the previous equation, we see that the distribution of the variable x n is determined by the distribution of the random variable Θ n = n−1 j=0w n σ n , withw = 1 − w. In the mathematical community, the distribution of Θ is known as Bernoulli convolutions [32,33,34,35,36,37,38,39]. In 1935 Jessen and Wintner [33] proved that for 0 <w < 1 the distribution of Θ is either absolutely continuous or singular with respect to Lebesgue measure.…”

Section: The Average Number Of Fermions In the System: N Nmentioning

confidence: 99%

“…Since then the challenge has been to identify the set ofw values for which the distribution is absolutely continuous or singular. Amongst the many noticeable results (see reviews [34,35]), P. Erdos [36] showed that the density is singular for allw ∈]1/2, 1[ such that 1/w is a Pisot number. No other than reciprocal Pisot numbers are known to be associated to a singular distribution [35].…”

Section: The Average Number Of Fermions In the System: N Nmentioning

confidence: 99%

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Quantum repeated interactions and the chaos game

Platini¹,

Low²

2018

J. Phys. A: Math. Theor.

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Inspired by the algorithm of Barnsley's chaos game, we construct an open quantum system model based on the repeated interaction process. We shown that the quantum dynamics of the appropriate fermionic/bosonic system (in interaction with an environment) provides a physical model of the chaos game. When considering fermionic operators, we follow the system's evolution by focusing on its reduced density matrix. The system is shown to be in a Gaussian state (at all time t) and the average number of particles is shown to obey the chaos game equation. Considering bosonic operators, with a system initially prepared in coherent states, the evolution of the system can be tracked by investigating the dynamics of the eigenvalues of the annihilation operator. This quantity is governed by a chaos game-like equation from which different scenarios emerge.

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Section: The Average Number Of Fermions In the System: N Nmentioning

confidence: 99%

Section: The Average Number Of Fermions In the System: N Nmentioning

confidence: 99%

Quantum repeated interactions and the chaos game

Platini¹,

Low²

2018

J. Phys. A: Math. Theor.

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“…These are particularly interesting being the only known singular Bernoulli convolutions, see [3,19,21]. There is a long history of studying the dimensionality properties of these measures, c.f., [17,22,23] and the many references cited therein for historical information. In [6,7], Feng conducted a study of mainly unbiased Bernoulli convolutions with contraction factor the inverse of a simple Pisot number and proved that for this class of measures the set of local dimensions is an interval.…”

Section: Introductionmentioning

confidence: 99%

Local Dimensions of Overlapping Self-Similar Measures

Hare¹

2019

Real Analysis Exchange

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We show that any equicontractive, self-similar measure arising from the IFS of contractions (S j ), with self-similar set [0, 1], admits an isolated point in its set of local dimensions provided the images of S j (0, 1) (suitably) overlap and the minimal probability is associated with one (resp., both) of the endpoint contractions. Examples include m-fold convolution products of Bernoulli convolutions or Cantor measures with contraction factor exceeding 1/(m + 1) in the biased case and 1/m in the unbiased case. We also obtain upper and lower bounds on the set of local dimensions for various Bernoulli convolutions.

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“…These have been extensively studied since the 1930's when Erdös [7] showed that if ̺ ∈ (1, 2) was a Pisot number, then the Bernoulli convolution was purely singular and later, in [8], that the Bernoulli convolutions were absolutely continuous for almost all ̺ ∈ (1,2). For more on the history of these classical problems see [19,21].…”

Section: Introductionmentioning

confidence: 99%