Related papers: Geometry in the Furstenberg Conjecture

We study atomic measures on $[0,1]$ which are invariant both under multiplication by $2\mod 1$ and by $3\mod 1$, since such measures play an important role in deciding Furstenberg's $\times 2, \times 3$ conjecture. Our specific focus was…

For each integer $n\ge 1$, denote by $T_{n}$ the map $x\mapsto nx\mod 1$ from the circle group $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ into itself. Let $p,q\ge 2$ be two multiplicatively independent integers. Using Baire Category arguments, we…

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…

In this paper, a polynomial version of Furstenberg joining is introduced and its structure is investigated. Particularly, it is shown that if all polynomials are non-linear, then almost every ergodic component of the joining is a direct…

We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p \mod 1$ and $\times q \mod 1$, respectively, and if $\log p/\log q\notin \mathbb{Q}$, then for all real numbers $u$ and…

We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed…

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to…

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a…

Motivated to generalize the Fourier frame concept to Banach spaces we introduce (p, q)-Bessel/frame measures for a given finite measure on LCA groups. We also present a general way of constructing (p, q)-Bessel/frame measures for a given…

We study some interesting properties of Furstenberg's topology of the integers. We show that it is metrizable, totally disconnected, and (Z,+,.) is a topological ring with respect to this topology. As an application, we show that any two…

For a probability measure $\mu$ on SL d (R), we consider the Furstenberg stationary measure on the space of flags. Under general non-degeneracy conditions, if $\mu$ is discrete and if g log g d$\mu$(g) < +$\infty$, then the measure $\nu$ is…

By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In…

We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers $p$ and $q$, we show that Furstenberg's $\times p,\times q$ conjecture holds if and only if…

Given a sequence of subsets A_n of {0,...,n-1}, the Furstenberg correspondence principle provides a shift-invariant measure on Cantor space that encodes combinatorial information about infinitely many of the A_n's. Here it is shown that…

We propose a counting dimension for subsets of Z and prove that, under certain conditions on two such subsets E and F, for Lebesgue almost every real \lambda\ the counting dimension of E+[\lambda F] is at least the minimum between 1 and the…

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration…

We construct Feigenbaum quadratic polynomials whose Julia sets have positive Lebesgue measure. They provide first examples of rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set. The…

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…

Furstenberg--Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional…

Let $\theta$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(\theta)$ is strongly irreducible and proximal. Let $\mu$ denote the Furstenberg…