Dynamical Systems and Analysis Seminar 2022 - 2023

In this talk we will consider the dynamical system given by the iteration of a rational map Q over the Riemann Sphere. The dynamics of Q split the Riemann Sphere into two totally invariant sets. The Fatou set consists of all points z such that the family of iterates of Q is normal, or equivalently equicontinuous, in some open neighbourhood of z. The Fatou set is open and corresponds to the set of points with stable dynamics. Its complement, the Julia set, is closed and corresponds to the set of points which present chaotic behaviour.

Fatou components, connected components of the Fatou set, are mapped amongst themselves under iteration of Q. A periodic Fatou component can only have connectivity 1, 2, or infinity. Despite that, preperiodic Fatou components can have arbitrarily large finite connectivity. There exist explicit examples of rational maps with Fatou components of any prescribed connectivity. However, the degree of these maps grows as the required connectivity increases.

We study a family of singular perturbations of rational maps with a single free critical point. Under certain conditions, the dynamical planes of these singular perturbations contain Fatou components of arbitrarily large finite connectivity. In this talk we will analyze the dynamical conditions under which these Fatou components of arbitrarily large connectivity appear.

Thurston's pseudo-Anosov homeomorphisms play a key role in the study of the dynamics of surface homeomorphisms. In this talk I'll introduce the more general measurable pseudo-Anosovs, discussing their dynamics and how they arise naturally from inverse limits of tent maps. This is joint work with Philip Boyland and André de Carvalho.

Furstenberg’s dynamical proof of the Szemerédi theorem initiated a thorough examination of multiple ergodic averages, laying the grounds for a new subfield within ergodic theory. Of special interest are averages of commuting transformations with polynomial iterates, which play a central role in Bergelson and Leibman’s proof of the polynomial Szemerédi theorem. Their norm convergence has been established in a celebrated paper of Walsh, but for a long time, little more has been known about them due to obstacles encountered by existing methods. Recently, there has been an outburst of research activity which sheds new light on their limiting behaviour. I will discuss a number of novel results, including new seminorm estimates and limit formulas for these averages. Additionally, I will talk about new criteria for joint ergodicity of general families of integer sequences whose potential utility reaches far beyond polynomial sequences. The talk will be based on recent papers written jointly with Nikos Frantzikinakis.

It is possible to define the topological entropy h_top(f) of a complex polynomial mapping f(x)= f(x_1, ... , x_n) := (P_1(x), ... , P_n(x)). It is a finite non-negative real number measuring the dynamical complexity of f. In one variable, h_top(f) equals the logarithm of the degree of f. I will discuss some results on the computation of h_top(f) in higher dimensions.

In this talk I will discuss rates of recurrence for Bernoulli measures defined on the full shift (more generally Gibbs measures defined on shifts of finite type). Our main result establishes a new critical threshold where the measure of the set of points satisfying a recurrence rate transitions from 0 to 1. This fact is an interesting consequence of the law of the iterated logarithm. This talk is based upon a joint work with Demi Allen and Balazs Barany.

A system of polynomial equations is called partition regular if every finite colouring of the positive integers produces monochromatic solutions to the system. A system is called density regular if it has solutions over every set of integers with positive upper density. Both of these properties are well understood for linear systems. Recently, there has been significant progress on characterising regularity for more general Diophantine systems, with ideas from dynamics, harmonic analysis, and analytic number theory being fruitfully applied to such problems. In this talk, I will discuss my recent work with Sam Chow on classifying density and partition regularity for arbitrary polynomial equations in sufficiently many variables.

Tom Kempton, Leticia Pardo-Simón and Richard Webb shared some open problems.

The Assouad dimension captures the coarse scaling behaviour of a set: it is the maximal Hausdorff dimension of a ``tangent'' of the set, i.e. a limit of zoomed-in images of the set. For sets with more structure (such as attractors of iterated function systems) one might hope to say more about its tangents.

A reasonable guess for a properly self-affine set $F\subset\R^2$ is that the Assouad dimension is the sum of the Assouad dimension of the projection $\pi(F)$ plus the maximal Assouad dimension of a slice $F\cap\pi^{-1}(x)$, for an appropriately chosen orthogonal projection $\pi$. In other words, the tangents look like products of the projection of a self-affine set with a slice orthogonal to the projection.

I will discuss this type of slicing formula in the special case of rectangular self-affine sets with principal projection satisfying a certain separation property, but otherwise with arbitrary overlaps. This work is joint with Jonathan Fraser.