Tag - Dynamical systems

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.

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.

Adi Glücksam: Unbounded Fast Escaping Wandering Domains

21st March, 2023

Complex dynamics explores the evolution of points under iteration of functions of complex variables. In this talk I will introduce into the context of complex dynamics, a new approximation tool allowing us to construct new examples of entire functions and show new possible dynamical behaviours. In particular, we answer a question of Rippon and Stallard from 2012 about unbounded wandering domains with unbounded orbits, and provide a collection of examples supporting a conjecture of Baker.

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.

Borys Kuca: Degree Lowering Along Arithmetic Progressions

6th March, 2023

Ever since Furstenberg proved his multiple recurrence theorem, the limiting behaviour of multiple ergodic averages along various sequences has been an important area of investigation in ergodic theory. In this talk, I will discuss averages along arithmetic progressions in which the differences are elements of a fixed integer sequence. Specifically, I will give necessary and sufficient conditions under which averages of fixed length of the aforementioned form have the same limit as averages along arithmetic progressions of the same length. The result relies on a higher-order version of the degree lowering argument, which is of independent interest. The talk is based on a joint work with Nikos Frantzikinakis.

Or Landesberg: Non-Rigidity of Horocycle Orbit Closures in Geometrically Infinite Surfaces

31st January, 2023

Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider ℤ-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.