Dynamical Systems and Analysis Seminar 2021 - 2022

We will talk about times 2,3,5 orbits on 2D-torus and show that irrational points have dense orbits. This work extends Furstenberg's celebrated times 2, times 3 theorem which says that times 2,3 orbits for irrationals are dense in 1D-torus.

Discussion of the paper "Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves" on work of Mirzakhani.

Discussion of the paper "Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves" on work of Mirzakhani.

I will discuss my recent work with Tuomas Orponen and Tuomas Sahlsten in which we prove that all compact subsets of the plane of almost full Hausdorff dimension contain a pattern (x, y), (x + t, y + t^2) for some nonzero t. Compared to previous works on similar topics, we do not require any assumption on the Fourier dimension of the set.

We will introduce in this talk renormalisation methods to describe the transition to chaos for simple Lorenz maps with plateaus. This work is associated with the study of open maps and global bifurcations and can find applications in fluid dynamics and electronics. A well-known and simple example is the standard unimodal map µx(x-1) where the transition to chaos happens with the creation of orbits of period 2n. The unimodal map has positive entropy when it has orbits of period 2n for all n. In this talk, we will focus on the doubling map, the tent map and two other similar functions with plateaus. We will present an induction process and describe its evolution to find all the possible routes to chaos for these four functions. This talk is based on joint work with Paul Glendinning.

In the study of the dynamics of a transcendental entire function f, we aim to describe its locus of chaotic behaviour, known as its Julia set and denoted by J(f). For many such f, the Julia set is a collection of unbounded curves that escape to infinity under iteration and form a Cantor bouquet, i.e., a subset of the complex plane ambiently homeomorphic to a straight brush. We show that there exists f whose Julia set J(f) is a collection of escaping curves, but J(f) is not a Cantor bouquet. On the other hand, we prove for certain f that if J(f) contains an absorbing Cantor bouquet, that is, a Cantor bouquet to which all escaping points are eventually mapped, then J(f) must be a Cantor bouquet. This is joint work with L. Rempe.

Badly approximable vectors are fractal sets enjoying rich Diophantine properties. In this respect, they play a crucial role in many problems well beyond Number Theory and Fractal Geometry (e.g., in signal processing, in mathematical physics and in convex geometry).

After outlining some of the latest developments in this very active area of research, we will take an interest in the Littlewood conjecture (c. 1930) and in its variants which all admit a natural formulation in terms of properties satisfied by badly approximable vectors. We will then show how ideas emerging from the mathematical theory of quasicrystals and from the theory of aperiodic tilings have recently enabled us to refute the so-called t-adic Littlewood conjecture.

All necessary concepts will be defined in the talk. Some of the results are joint with Fred Lunnon (Maynooth) and Erez Nesharim (Hebrew University of Jerusalem).

In this talk we'll consider a class of Bernoulli measures in the plane, which are supported on attractors of iterated function systems consisting of nonlinear, non-conformal maps. These maps have triangular Jacobian matrices and satisfy an appropriate separation condition. Using ideas from thermodynamic formalism we'll see how the L^q-spectrum of such measures can be calculated, and as a corollary obtain the box dimension of the sets they are supported on. This is joint work with Kenneth Falconer and Jonathan Fraser.