Tag - Ergodic theory

Stanisław Kasjan: On Dynamics of ℬ-free Systems Generated by Behrend Sets

6th June, 2023

Given a set ℬ of natural numbers, not containing 1, we denote by ℱℬ the set of ℬ-free numbers, that is, ℱℬ = ℤ \ ⋃b∈ℬ bℤ. Let Xη be the ℬ-free subshifts, that is the subshift induced by η, where η denotes the characteristic function of ℱℬ . That means, Xη is the closure of the set of all shifts of η in the space {0, 1}ℤ equipped with the product topology. We are interested in the case when B is a Behrend set, that is, when the set of ℬ-free numbers has zero density. It turns out that this is the case precisely when Xη is proximal and has zero entropy. We prove that the complexity of Xη, with ℬ being a Behrend set, can achieve any subexponential growth. Together with the ℬ-free shift we investigate the ℬ-admissible shift Xℬ and we show that it is transitive if and only if the set B is pairwise coprime, which allows one to characterize dynamically the subshifts generated by the Erdős sets (infinite, coprime and not Behrend). We also estimate the complexity for some classical subshifts (the subshift of primes or semi-primes). The lower estimates are obtained conditionally on Hardy-Littlewood Conjecture or Dickson’s Conjecture. We remark on a recent result of Tao and Ziegler (not assuming the conjectures) that the shift of primes is uncountable.

Satvik Singh: Diagonal unitary covariant quantum channels

22nd May, 2023

The talk will present a study of (finite-dimensional) quantum channels which are covariant under the action of the diagonal unitary group. Many salient examples, such as the depolarizing channels, dephasing channels, amplitude damping channels, and mixtures thereof, lie in this class. The first part of the talk will be devoted to the study of entanglement properties of these channels. In particular, by reformulating the entanglement-breaking property of such channels in terms of the cone of pairwise completely positive matrices, I will show that the well-known PPT-squared conjecture holds for channels in this class. I will also unravel an interesting connection between the entanglement-breaking property of such channels and triangle-free graphs. The second half of the talk will deal with the ergodic properties of these channels. I will show that the ergodic behaviour of a channel in this class is essentially governed by a classical stochastic matrix, thus allowing us to exploit tools from classical ergodic theory to study quantum ergodicity of such channels.

Peter Humphries: Restricted Arithmetic Quantum Unique Ergodicity

4th May, 2023

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I will discuss a variant of this problem in this arithmetic setting concerning the mass equidistribution of Hecke-Maass cusp forms on submanifolds of the modular surface.

Ethan Ackelsberg: Polynomial Patterns in Finite Fields: a Dynamical Point of View

18th April, 2023

In its dynamical formulation, the Furstenberg-Sárközy theorem states that for any invertible measure-preserving system (X,μ,T), any set A⊆X with μ(A)>0, and any integer polynomial P with P(0)=0,

c(A)=limN−M→∞ 1/(N−M) ∑n=MN−1 μ(A∩TP(n)A)>0.

The limit c(A) obtains the 'correct' value μ(A)2 when T is totally ergodic. In fact, when T is totally ergodic, one has an ergodic theorem for polynomial actions: for any integer polynomial P and any f∈L2(μ),

limN−M→∞ 1/(N−M) ∑n=MN−1 TP(n)f= ∫X f dμ,

where the limit is taken in L2(μ). We will explain that the correct notion of total ergodicity for polynomial actions of more general rings depends on the dynamical behavior of actions along finite index ideals. From this point of view, the action of a large finite field on itself is asymptotically totally ergodic, since the index of the only proper ideal {0} grows with the size of the field. Guided by ergodic-theoretic results about polynomial (multiple) recurrence in totally ergodic systems, we then obtain several new results about polynomial configurations in large subsets of finite fields.

Amol Aggarwal: Universality results in random tiling models II

7th April, 2023

Random tilings of large domains have long been objects of fascination for mathematicians and physicists. What one witnesses for these systems is that the shapes of their domains have a significant impact on the local geometry of the tiling. Indeed, these shapes influence the probabilities of (microscopic) tiling patterns well inside the domain; can induce phase transitions (arctic curves) at facet edges; and can create "turning points" where the limiting behaviour of the model is discontinuous. At least for certain special shapes, there now exist powerful algebraic frameworks providing exact formulas, which can be analysed to precisely identify the limiting statistics that appear in each of these situations. This gives rise to a universality prediction, that the same limiting statistical behaviours should also arise in random tilings not only of those particular domains, but also of far more general ones. Over the past several years, there has been progress in establishing such universality statements for random tilings of the triangular lattice, or lozenge tilings. In these talks, we review some of these developments, and outline how they are obtained, by intertwining the above mentioned algebraic frameworks with analytic and probabilistic ones.

Amol Aggarwal: Universality results in random tiling models I

7th April, 2023

Random tilings of large domains have long been objects of fascination for mathematicians and physicists. What one witnesses for these systems is that the shapes of their domains have a significant impact on the local geometry of the tiling. Indeed, these shapes influence the probabilities of (microscopic) tiling patterns well inside the domain; can induce phase transitions (arctic curves) at facet edges; and can create "turning points" where the limiting behaviour of the model is discontinuous. At least for certain special shapes, there now exist powerful algebraic frameworks providing exact formulas, which can be analysed to precisely identify the limiting statistics that appear in each of these situations. This gives rise to a universality prediction, that the same limiting statistical behaviours should also arise in random tilings not only of those particular domains, but also of far more general ones. Over the past several years, there has been progress in establishing such universality statements for random tilings of the triangular lattice, or lozenge tilings. In these talks, we review some of these developments, and outline how they are obtained, by intertwining the above mentioned algebraic frameworks with analytic and probabilistic ones.

Rigoberto Zelada: Polynomial Ergodic Theorems for Strongly Mixing Commuting Transformations

4th April, 2023

The goal of this talk is to present new results dealing with the asymptotic joint independence properties of commuting strongly mixing transformations along polynomials. These results form natural strongly mixing counterparts to various weakly and mildly mixing  polynomial ergodic theorems. A decisive role in the proofs is played by a new notion of convergence that is adequate for dealing with strong mixing and, as we will see, cannot be avoided while working with commuting polynomial actions. This talk is based on joint work with Vitaly Bergelson.

Henrik Kreidler: Geometric Representation of Structured Extensions in Ergodic Theory

14th March, 2023

The Mackey-Zimmer representation theorem is a key structural result from ergodic theory: Every compact extension between ergodic measure-preserving systems can be written as a skew-product by a homogeneous space of a compact group. This is used, e.g., in Furstenberg's original ergodic theoretic proof of Szemerédi's theorem, as well as in the classical proofs of the Host-Kra-Ziegler structure theorem for characteristic factors. Inspired by earlier work of Ellis, we discuss a topological approach, first to the original theorem, and then to a generalization relaxing the ergodicity assumptions due to Austin.

Terence Tao: Almost all Collatz orbits attain almost bounded values

13th March, 2023

Define the Collatz map Col on the natural numbers by setting Col(n) to equal 3n+1 when n is odd and n/2 when n is even. The notorious Collatz conjecture asserts that all orbits of this map eventually attain the value 1. This remains open, even if one is willing to work with almost all orbits rather than all orbits. We show that almost all orbits n, Col(n), Col2(n), ... eventually attain a value less than f(n), for any function f that goes to infinity (no matter how slowly). A key step is to obtain an approximately invariant (or more precisely, self-similar) measure for the (accelerated) Collatz dynamics.