The paper investigates quantitative weak mixing of Salem substitutions flows. We prove that for a substitution whose substitution matrix is irreducible over the rationals and the dominant eigenvalue is a Salem number, for almost every suspension flow with a piecewise constant roof function, quantitative weak mixing holds with a rate that is slightly worse than a power of loglog. We do not know if this is sharp, but we do show that for any suspension flow of this kind, quantitative weak mixing with a polynomial rate is impossible. Results for specific systems are often much weaker than for “typical” or “generic” ones. In the Appendix we explain how a minor modification of an argument from Bufetov and Solomyak (2014) yields very weak, but nevertheless quantitative weak mixing estimates of log^* type for the self-similar suspension flow over a Salem substitution. Simultaneously this provides first quantitative decay rates for the Fourier transform of Salem Bernoulli convolutions.

This paper is partly an exposition, and partly an extension of our work [Adv. Math. 399 (2022)] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of loc. cit. and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS’s with place-dependent probabilities and natural (equilibrium) measures for smooth IFS’s. One of the goals of this paper is to present an exposition of loc. cit. in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by Károly Simon at the Summer School “Dynamics and Fractals” in 2023 at the Banach Center, Warsaw. The main new feature, compared to loc. cit., is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures.

The paper is devoted to the properties of a complex matrix “twisted,” otherwise called “spectral,” cocycle, associated with substitution dynamical systems. Following a recent finding of Rajabzadeh and Safaee [arXiv:2501.16824] of an invariant section for the twisted cocycle, we indicate that this implies presence of a zero Lyapunov exponent. This has consequences for the spectral properties of substitution dynamical systems; in particular, this extends the scope and simplifies the proof of singular spectrum for a large class of substitutions on two symbols. We also obtain some results on positivity of the top exponent. In the appendix we compute the Lebesgue almost everywhere local dimension of spectral measures of some “simple” test functions, for almost every irrational rotation. This sheds some light on the earlier work of Bufetov and the author, relating the local dimension of spectral measures to pointwise Lyapunov exponents of the twisted cocycle. It should be noted that the paper has some (mutually acknowledged) overlap with [arXiv:2501.16824, Appendix].

Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = \lambda_i z + t_i,\ \lambda_i, t_i \in \mathbb{C},\ |\lambda_i|<1, i=1,\ldots, k$. We prove that for almost every choice of $(\lambda_1, \ldots, \lambda_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest.

We consider iterated function systems (IFS) in ℝ^d for d≥ 3 of the form {f_j(x) = λ𝒪 x + a_j}_j=0^m, with a_0=0 and m≥ 1. Here λ∈ (0,1) is the contraction ratio and 𝒪 is an orthogonal matrix. Given a positive probability vector p, there is a unique invariant (stationary) measure for the IFS, called (in this case) a homogeneous self-similar measure, which we denote μ(λ𝒪, 𝒟, p), where 𝒟 = {a_0,…,a_m} is the set of “vector digits”. We obtain two results on Fourier decay for such measures. First we show that if 𝒟 spans ℝ^d, then for every fixed 𝒪 and p the measure μ(λ𝒪, 𝒟, p) has power Fourier decay (equivalently, positive Fourier dimension) for all but a zero-Hausdorff dimension set of λ. In our second result we do not impose any restrictions on 𝒟, other than the necessary one of affine irreducibility, and obtain power Fourier decay for almost all homogeneous self-similar measures; however, only for even d≥ 4. Combined with recent work of Corso and Shmerkin [arXiv:2409.04608] , these results imply absolute continuity for almost all self-similar measures under the same assumptions, in the super-critical parameter region.

The paper investigates H\"older and log-H\"older regularity of spectral measures for weakly mixing substitutions and the related question of quantitative weak mixing. It is assumed that the substitution is primitive, aperiodic, and its substitution matrix is irreducible over the rationals. In the case when there are no eigenvalues of the substitution matrix on the unit circle, our main theorem says that a weakly mixing substitution $\mathbb{Z}$-action has uniformly log-H\"older regular spectral measures, and hence admits power-logarithmic bounds for the rate of weak mixing. In the more delicate Salem substitution case, our second main result says that H\"older regularity holds for algebraic spectral parameters, but the H\"older exponent cannot be chosen uniformly.

The paper is concerned with random S-adic systems arising from an i.i.d. sequence of unimodular substitutions. Using equidistribution results of Benoist and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the Lyapunov exponent of the spectral cocycle is strictly less that 1/2 of the Lyapunov exponent of the random walk on SL(2,ℝ) driven by the sequence of substitution matrices, then almost surely the spectrum of the S-adic ℤ-action is singular with respect to any (fixed in advance) continuous measure. Finally, the appendix (by Pascal Hubert and Carlos Matheus) discusses the weak-mixing property for random S-adic systems associated to the family of substitutions introduced in Example 4.1.

Abstract The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math. 141(1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$ , allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following the work of Treviño [Quantitative weak mixing for random substitution tilings. Israel J. Math., to appear], in the simpler, non-random setting. We review some of the results of Treviño in this special case and illustrate them on concrete examples.

This is a brief survey of selected results obtained using the “transversality method” developed for studying parametrized families of fractal sets and measures. We mostly focus on the early development of the theory, restricting ourselves to self-similar and self-conformal iterated function systems.

This survey of the spectral properties of substitution dynamical systems is devoted to primitive aperiodic substitutions and associated dynamical systems: Z \mathbb {Z} -actions and R \mathbb {R} -actions, the latter viewed as tiling flows. The focus is on the continuous part of the spectrum. For Z \mathbb {Z} -actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. References are given to complete proofs and emphasize ideas and various links.

A sufficient condition for a substitution automorphism to have pure singular spectrum is given in terms of the top Lyapunov exponent of the associated spectral cocycle. As a corollary, singularity of the spectrum is established for an infinite family of self-similar interval exchange transformations.

We consider one-parameter families of smooth uniformly contractive iterated function systems {fjλ} on the real line. Given a family of parameter dependent measures {μλ} on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps Πλ. The main novelty of our work is that the measures μλ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure (Πλ)⁎μλ, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if μλ are Gibbs measures for a family of Hölder continuous potentials ϕλ, with Hölder continuous dependence on λ and {Πλ} satisfy the transversality condition, then the projected measure (Πλ)⁎μλ is absolutely continuous for Lebesgue a.e. λ, such that the ratio of entropy over the Lyapunov exponent is strictly greater than 1. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).

The paper corrects the formulation of [3, Prop. 5.4], whose proof is unchanged.

We show that for Lebesgue almost all d -tuples (\theta_1,\ldots,\theta_d) , with |\theta_j|>1 , any self-affine measure for a homogeneous non-degenerate iterated function system \{Ax+a_j\}_{j=1}^m in \mathbb{R}^d , where A^{-1} is a diagonal matrix with the entries (\theta_1,\ldots,\theta_d) , has power Fourier decay at infinity.

The construction of spectral cocycle from the case of 1-dimensional substitution flows by Bufetov-Solomyak [arXiv:1802.04783] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$, allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following Trevi\~no [arXiv:2006.16980], in the simpler, non-random setting. We review some of the results on quantitative weak mixing from [arXiv:2006.16980] in this special case and illustrate them on concrete examples.

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the H\"older property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained H\"older estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove H\"older regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.

We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({\mathbb{R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({\mathbb{R}})$ matrices induces a (generalized) iterated function system on the projective line ${\mathbb{RP}}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erdős and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.