We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $λ$ in the unit disk such that the attractor $A_λ$ of the IFS $\{λz-1, λz+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_λ$ themselves and to natural measures $ν_λ$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erdös and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $ν_λ$. Next we investigate the Hausdorff dimension and measure of $A_λ$, for $λ$ in the set $M$, for Lebesgue-a.e. $λ$. We also obtain partial results on the absolute continuity of $ν_λ$ for a.e. $λ$ of modulus greater than $\sqrt{1/2}$.
It is well known that the Bernoulli convolution $ν_β$ associated to the golden mean has Hausdorff dimension less than 1, i.e. that there exists a set $A$ with $ν_β(A)=1$ and $dim_H(A)<1$. We construct such a set $A$ explicitly and discuss how our approach might be generalised to prove the singularity of other Bernoulli convolutions
We describe a family $ϕ_λ$ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems $ϕ_λ$ and give some numerical evidence to suggest values of $λ$ for which $ϕ_λ$ may be piecewise convex.
We identify a family of numbers for which the Bernoulli convolution is singular. Within this family we find two countable collections of Salem numbers in the interval $(1,2)$, and another Salem number and an algebraic integer that is neither Pisot nor Salem in $(1,2)$. It also contains a non-Pisot, non-Salem algebraic number bigger than 3. Hence, we provide the first new explicit examples of singular Bernoulli convolutions since the work of Erdős in 1939.
Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinatorial structure. Visualizing this structure we discuss results of Erdös, Jóo, Komornik, Sidorov, de Vries, Jordan, Shmerkin and Solomyak, Feng and Wang. We emphasize the rôle of finite orbits of associated multivalued maps and mention a few new properties and examples.
Bernoulli convolutions are certain measures on the unit interval depending on a parameter $β$ between 1 and 2. In spite of their simple definition, they are not yet well understood. We study their two-dimensional density which exists by a theorem of Solomyak. To each Bernoulli convolution, there is an interval $D$ called the overlap region, and a map which assigns two values to each point of $D$ and one value to all other points of $[0,1].$ There are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density, respectively. Orbits w…
We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the $β$-numeration. A matrix decomposition of these measures is obtained in the case when $β$ is a PV number. We also determine their Gibbs properties for $β$ being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.
We study the typical growth rate of the number of words of length n which can be extended to beta-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter beta is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $μ_λ$ is the probability measure corresponding to the law of the random variable $ξ= \sum_{k=0}^\infty ξ_kλ^k$, where $ξ_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < λ< 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(μ_λ) \geq 0.96399$ for all …
We prove that the set of exceptional $λ\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.
Let $λ\in (1,\sqrt{2}]$ be an algebraic integer with Mahler measure $2.$ A classical result of Garsia shows that the Bernoulli convolution $μ_λ$ is absolutely continuous with respect to the Lebesgue measure with a density function in $L^\infty$. In this paper, we show that the density function is continuous.
We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrized families of self-similar sets and measures in the complex plane, extending earlier results.
In [17] the author and A. Vershik have shown that for $\be=\frac12(1+\sqrt5)$ and the alphabet $\{0,1\}$ the infinite Bernoulli convolution ($=$ the Erdös measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type $\mathrm{II}_1$ under the $\be$-shift, and the natural extension of the $\be$-shift provided with the measure equivalent to the Erdös measure, is Bernoulli. In this note we extend this result to all Pisot parameters $\be$ (modulo some general arithmetic conjecture) and an arbitrary "sufficient" alphabet.
We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $ν_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain a…
We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.
The Bernoulli convolution with parameter $λ\in(0,1)$ is the measure on $\bf R$ that is the distribution of the random power series $\sum\pmλ^n$, where $\pm$ are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.
We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $ν_β$ to arbitrary given accuracy whenever $β$ is algebraic. In particular, if the Garsia entropy $H(β)$ is not equal to $\log(β)$ then we have a finite time algorithm to determine whether or not $\mathrm{dim}_\mathrm{H} (ν_β)=1$.
We study the signed Bernoulli convolution $$ν_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}-\frac12δ_{-β^{-j}}\right ),\ n\ge 1$$ where $β>1$ satisfies $$β^m=β^{m-1}+\cdots+β+1$$ for some integer $m\ge 2$. When $m$ is odd, we show that the variation $|ν_β^{(n)}|$ coincides the unsigned Bernoulli convolution $$μ_β^{(n)}=*_{j=1}^n \left (\frac12δ_{β^{-j}}+\frac12δ_{-β^{-j}}\right ).$$ When $m$ is even, we obtain the exact asymptotic of the total variation $\|ν_β^{(n)}\|$ as $n\rightarrow\infty$.
Let $β\in(1,2)$ and let $H_β$ denote Garsia's entropy for the Bernoulli convolution $μ_β$ associated with $β$. In the present paper we show that $H_β>0.82$ for all $β\in (1, 2)$ and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields $\dim (μ_β)\ge0.82$ for all $β\in(1,2)$. In addition, we show that if an algebraic $β$ is such that $[\mathbb{Q}(β): \mathbb{Q}(β^k)] = k$ for some $k \geq 2$, then $\dim(μ_β)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.
The random beta-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special values of beta and use the projection to obtain results on the local dimension of the Bernoulli convolution.
The Bernoulli convolution $ν_λ$ with parameter $λ\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pmλ^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\dimν_λ=1$ for all transcendental $λ\in(1/2,1)$.
Let $β\in(1,2)$ be a Pisot number and let $H_β$ denote Garsia's entropy for the Bernoulli convolution associated with $β$. Garsia, in 1963 showed that $H_β<1$ for any Pisot $β$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_β$ are known. In the present paper we show that $H_β>0.81$ for all Pisot $β$, and improve this lower bound for certain ranges of $β$. Our me…
Let $ν_λ^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n λ^n$, where $i_n$ is a sequence of i.i.d. random variables taking the values 0,1 with probabilities $p,1-p$. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of $ν_λ^p$ for typical $λ$. Namely, we investigate the size of the sets \[ Δ_{λ,p}(α) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log ν_λ^p(B(x,r))}{\log r} =α\right\}. \] Our main results highlight the fact that for almost all, and in some cases all, $λ$ in an appropriate range, $Δ_{λ,p}(α)$ is non…
In this paper we give an example of uniform convergence of the sequence of column vectors $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$, $A_i\in\{A,B,C\}$, $A,B,C$ being some $(0,1)$-matrices of order $7$ with much null entries, and $V$ a fixed positive column vector. These matrices come from the study of the Bernoulli convolution in the base $β>1$ such that $β^3=2β^2-β+1$, that is, the (continuous singular) probability distribution of the random variable $\displaystyle(β-1)\sum_{n=1}^\infty{ω_n\overβ^n}$ when the independent random variables $ω_n$ take the values $0$ …
It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $μ_θ$ parameterized by a Pisot number $θ$, is countable. Combined with results of Salem and Sarnak, this proves that for every fixed $θ>1$ the spectrum of the convolution operator $f\mapsto μ_θ*f$ in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all $p\in(1,\infty)$, namely, $\bar{\{\hat{μ_θ}(n) : n\in\mathbb{Z}\}}$. Our result answers the question raised by P. Sarnak in \cite{Sar}. We also consider the sets $\bar{\{\hat{μ_θ}(rn) : n\in\mathbb{Z}\}}$ for $r>0$ which corr…
We examine measure preserving mappings $f$ acting from a probability space $(Ω, F,μ) $ into a probability space $% (Ω^{*},F^{*},μ^{*}) ,$ where $μ^{*}=μ(f^{-1})$. Conditions on $f$, under which $f$ preserves the relations ''to be singular'' and ''to be absolutely continuous'' between measures defined on $(Ω, F) $ and corresponding image measures, are investigated. We apply the results to investigate the distribution of the random variable $% ξ=\sum\limits^{\infty}_{k=1} ξ_kλ^k,$ where $% λ\in (0;1),$ and $ξ_k$ are independent not necessarily identically distributed random variables taking the …
Max-semistable extremal behavior of AR(1)-processes connected with Bernoulli convolutions
Mathematical Institute, Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学数学研究所）；Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen（格罗宁根大学伯努利数学、计算机与人工智能研究所）
We consider the multifractal structure of the Bernoulli convolution $ν_λ$, where $λ^{-1}$ is a Salem number in $(1,2)$. Let $τ(q)$ denote the $L^q$ spectrum of $ν_λ$. We show that if $α\in [τ'(+\infty), τ'(0+)]$, then the level set $$E(α):={x\in \R:\; \lim_{r\to 0}\frac{\log ν_λ([x-r, x+r])}{\log r}=α}$$ is non-empty and $\dim_HE(α)=τ^*(α)$, where $τ^*$ denotes the Legendre transform of $τ$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[τ'(+\infty), τ'(0+)]$ is not a singleton when $λ^{-1}$ is the la…
The Bernoulli convolution with parameter $λ\in(0,1)$ is the probability measure $μ_λ$ that is the law of the random variable $\sum_{n\ge0}\pmλ^n$, where the signs are independent unbiased coin tosses. We prove that each parameter $λ\in(1/2,1)$ with $\dimμ_λ<1$ can be approximated by algebraic parameters $ξ\in(1/2,1)$ within an error of order $\exp(-deg(ξ)^{A})$ for any number $A$, such that $\dimμ_ξ<1$. As a corollary, we conclude that $\dimμ_λ=1$ for each of $λ=\ln 2, e^{-1/2}, π/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conje…
We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar measure. In the special case of Bernoulli convolutions this paper gives a sufficient condition for the Bernoulli convolution with parameter lambda to be absolutely continuous in terms of lambda and its Mahler measure.
Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss the local structure of this measure. We also make links with the question of the Hausdorff dimension of the corresponding Bernoulli Convolution
We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continui…
We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with $m$-fold sums of Cantor sets with ratio of dissection $1/R$ for integer $R\leq m$. We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in …
For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erd{ö}s [2] and Salem [12]. We prove weak forms of their results for general self-similar measures associated to affine contractions of the real line.
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have a…
We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and Bárány, Käenmäki. To overcome the difficulties stemming form these, we develop novel techniques, including an extension of Hochman's entropy increase method to a function field setup.
We present a self-contained proof of a formula for the $L^q$ dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the $L^q$ norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernou…
In this paper we study the set of digit frequencies that are realised by elements of the set of $β$-expansions. The main result of this paper demonstrates that as $β$ approaches $1,$ the set of digit frequencies that occur amongst the set of $β$-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.
The Bernoulli convolution associated to the real $β>1$ and the probability vector $(p_0,..,p_{d-1})$ is a probability measure $η_{β,p}$ on $\mathbb R$, solution of the self-similarity relation $\displaystyleη=\sum_{k=0}^{d-1}p_k\cdotη\circ S_k$ where $S_k(x)=\frac{x+k}β$. If $β$ is an integer or a Pisot algebraic number with finite Rényi expansion, $η_{β,p}$ is sofic and a Markov chain is naturally associated. If $β=b\in\mathbb N$ and $p_0=...=p_{d-1}=\frac1d$, the study of $η_{b,p}$ is close to the study of the order of growth of the number of representations in base $b$ with digits in $\{0,1…
