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 show that every totally ergodic generalised matrix equilibrium state is psi-mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.
We prove that a shift ergodic measure on a topologically mixing sub-shift is isomorphic to a Bernoulli shift whenever it is quasi invariant under permutations of finite number of coordinates. We prove also that Gibbs measures on topologically mixing subshift of finite type are quasi invariant.
{\abstract{\textwidth=4,5 in} A discrete time process, with law $μ$, is quasi-exchangeable if for any finite permutation $σ$ of time indices, the law $μ_σ$ of the resulting process is equivalent to $μ$. For a quasi-exchangeable stationary process we prove mainly (1) that if the process is ergodic then it is isomorphic to a Bernoulli process and (2) that if the family of all Radon-Nikodym derivatives $\{{dμ_σ\over dμ}\}$ is uniformly integrable then the process is a mixture of Bernoulli processes, which generalizes De Finetti's Theorem. We give application of (1) to some determinantal processes…
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.
Let $Γ$ be a countable group and denote by $\Cal S$ the equivalence relation induced by the Bernoulli action $Γ\curvearrowright [0,1]^Γ$, where $[0,1]^Γ$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\Cal R$ of $\Cal S$, there exists a partition $\{X_i\}_{i\geq 0}$ of $[0,1]^Γ$ with $\Cal R$-invariant measurable sets such that $\Cal R_{|X_0}$ is hyperfinite and $\Cal R_{|X_i}$ is strongly ergodic (hence ergodic), for every $i\geq 1$.
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.
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 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…
This investigation studies the ergodic properties of reversible linear cellular automata over $\mathbb{Z}_m$ for $m \in \mathbb{N}$. We show that a reversible linear cellular automaton is either a Bernoulli automorphism or non-ergodic. This gives an affirmative answer to an open problem proposed in [Pivato, Ergodc theory of cellular automata, Encyclopedia of Complexity and Systems Science, 2009, pp.~2980-3015] for the case of reversible linear cellular automata.
Previous work showed that all Bernoulli shifts over a free group are orbit-equivalent. This result is strengthened here by replacing Bernoulli shifts with the wider class of properly ergodic countable state Markov chains over a free group. A list of related open problems is provided.
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…
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 give elementary constructions of factors of nonsingular Bernoulli shifts. In particular, we show that all nonsingular Bernoulli shifts on a finite number of symbols which satisfy the Doeblin condition have a factor that is equivalent to an independent and identically distributed system. We also prove that there are type-III:1 Bernoulli shifts of every possible ergodic index, answering a question of Danilenko and Lemanczyk (Ergodic Theory Dynam. Systems, 39(12):3292-3321, 2019).
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 answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.
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.
We show that for any countable group, any free probability measure preserving action of the group weakly contains all Bernoulli actions of the group. It follows that for a finitely generated groups, the cost is maximal on Bernoulli actions and that all free factors of i.i.d.-s the group have the same cost. We also show that if a probability measure preserving action f is ergodic, but not strongly ergodic, then f is weakly equivalent to f\timesI where I denotes the trivial action on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy.
Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(μ_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}μ_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,μ_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for n…
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.
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin ent…
Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
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.
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.
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 investigate Gaussian actions through the study of their crossed-product von Neumann algebra. The motivational result is Chifan and Ioana's ergodic decomposition theorem for Bernoulli actions (Ergodic subequivalence relations induced by a Bernoulli action, {\it Geometric and Functional Analysis}{\bf 20} (2010), 53-67) that we generalize to Gaussian actions. We also give general structural results that allow us to get a more accurate result at the level of von Neumann algebras. More precisely, for a large class of Gaussian actions $Γ\curvearrowright X$, we show that any subfactor $N$ of $L^\i…
We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.
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 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…
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 $n\rightarrow\infty$.
We prove that a Lebesgue measure-preserving linked-twist map defined in the plane is metrically isomorphic to a Bernoulli shift (and thus strongly mixing). This is the first such result for an explicitly defined linked-twist map on a manifold other than the two-torus. Our work builds on that of Wojtkowski who established an ergodic partition for this example using an invariant cone-field in the tangent space.
We study the ergodic properties of a class of measures on $Σ^{\mathbb{Z}}$ for which $μ_{\mathcal{A},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\left \|A_{x_{0}}\cdots A_{x_{n-1}}\right \| ^{t}$, where $\mathcal{A}=(A_{0}, \ldots , A_{M-1})$ is a collection of matrices. The measure $μ_{\mathcal{A},t}$ is called a matrix Gibbs state. In particular we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques to understand these measures including a novel approach based on Perron-Frobenius theory. We find that when $t$ is an even integer…
In this paper, we study ergodic properties of hyperbolic measures with local product structure. We show that all the classical results that hold in the case of SRB measure hold for these measures. In particular, we show the decomposition in countably many ergodic components, we prove the decomposition into K-components, and show that for hyperbolic measure with local product structure, The K property implies the Bernoulli property. We also give some examples of measures where the results are applicable.
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…
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 $β\in(1,2)$ 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.
We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume pre…
In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called cylindric scatterers) have been removed. We prove that every such system is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for the ergodicity is present.
We provide a purely topological characterisation of uniquely ergodic topological dynamical systems (TDSs) whose unique invariant measure is zero entropy loosely Bernoulli (following Ratner, we call such measures loosely Kronecker). At the heart of our proofs lies Feldman-Katok continuity (FK-continuity for short), that is, continuity with respect to the change of metric to the Feldman-Katok pseudometric. Feldman-Katok pseudometric is a topological analog of f-bar (edit) metric for symbolic systems. We also study an opposite of FK-continuity, coined FK-sensitivity. We obtain a version of Auslan…
We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on $\Bbb T^2$ which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy …
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…
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 …
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 $0$ …
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 show that linear complexity is the threshold for the emergence of Kakutani inequivalence for measurable systems supported on a minimal subshift. In particular, we show that there are minimal subshifts of arbitrarily low super-linear complexity that admit both loosely Bernoulli and non-loosely Bernoulli ergodic measures and that no minimal subshift with linear complexity can admit inequivalent measures.
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center…
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）
