This is a three-day workshop on deterministic extremes and recurrence in dynamical systems. This meeting will take place over Zoom, and access links/codes will be provided on this site closer to the time of the meeting (see below).
Contact Mark Holland, Surabhi Desai, Peter Ashwin, and Tomas Persson if you are interested and come back to here for more information in due course. Details are as follows.
Links to notes/slides from talks:
Borel Cantelli notes (Persson)
Shrinking/moving targets (Koivusalo)
Extremes/maxima notes (Holland)
Extremes/energy-observables (Carney)
Clustering/extremes (Todd)
Attractors/measures (Newman)
Monday 21st June
Theme: Shrinking targets & recurrence
13:45 | Meeting opens | |
13.50 | Introduction | |
14:00-14.45 | Tomas Persson (Lund): Dynamical Borel–Cantelli lemmata, shrinking targets and recurrence | |
15:00-15.50 | Henna Koivusalo (Bristol): Path-dependent, shrinking, moving targets and beyond, on generic self-affine sets | |
16:15-17:00 | Tomas Persson. (2nd lecture). | |
17:00-17:30 | Informal discussion |
Tuesday 22nd June
Theme: Extremes, recurrence and limit laws in dynamical systems
13:00-13.45 | Mark Holland (Exeter): On the distribution of extreme events for dynamical systems | |
14:00-14.45 | Mark Holland. (2nd lecture). | |
15:00-15.50 | Meagan Carney (University of Queensland): Extremes for Energy-like Observables on Hyperbolic Systems. | |
16:15-17.05 | Mike Todd (St Andrews): Capturing clustering in extreme values | |
17:05-17:30 | Informal discussion |
Wednesday 23rd June
Theme: Extremes and applications
13:30-14.15 | Tobias Kuna (Reading): A qualitative aspect of extreme value theory for dynamical systems | |
14:30-15.15 | Tobias Kuna. (2nd lecture). | |
15.45-16.35 | Julian Newman (Exeter): Attractors and attracting measures | |
16:35-17:00 | Informal discussion |
The 2×45 min sessions are mini-courses accessible to PhD students/post-doc researchers. The 50 minute talks are on general current research in the relevant themes.
Abstracts
Meagan Carney (University of Queensland)
Extremes for Energy-like Observables on Hyperbolic Systems.
We consider an ergodic, measure-preserving dynamical system $(T, X, \mu)$ equipped with an observable $\phi: X\rightarrow R$. Given the stochastic process $X_n(x) = \phi(T^n(x))$, we establish an extreme value law for the sequence of maxima $M_n = max_{k\le n} X_k$ where $\phi$ is an energy-like observable and $(T, X, \mu)$ is hyperbolic. Observables of this form have the property that the set of maximization is a curve rather than a single point. We will discuss results in the case of Anosov diffeomorphisms, Sinai dispersing billiards, and coupled expanding maps. We will highlight the dependence of the extremal index on the set of maximization and discuss some numerical results for these systems.
Mark Holland (Exeter)
On the distribution of extreme events for dynamical systems
This lecture will cover almost sure growth bounds for maxima (extremes), using links to the theory of dynamical Borel Cantelli Lemmas. We also review distributional convergence results of extremes for dynamical systems, contrasting to the classical theory of extremes for i.i.d random variables.
Henna Koivusalo (Bristol)
Path-dependent, shrinking, moving targets and beyond, on generic self-affine sets
The classical shrinking target problem concerns the following set-up: Given a dynamical system (T, X) and a sequence of targets (B_n) of X, we investigate the size of the set of points x of X for which T^n(x) hits the target B_n for infinitely many n. In this talk I will study shrinking target problems in the context of fractal geometry. I will first recall the symbolic and geometric dynamical systems associated with iterated function systems, fundamental constructions from fractal geometry. I will then briefly cover the Hausdorff dimension theory of generic self-affine sets; that is, sets invariant under affine iterated function systems with generic translations. Finally, I will show how to calculate the Hausdorff dimension of shrinking target-type sets on generic self-affine sets. The target sets that I will consider move and shrink at a speed that depends on the path of x. Time permitting, further problems of similar flavour and refinements of the dimension result might also be explored. This talk is based on a joint work with Lingmin Liao and Michal Rams.
Tobias Kuna (Reading)
A qualitative aspect of extreme value theory for dynamical systems
Julian Newman (Exeter)
Attractors and attracting measures
Under mild assumptions, the SRB measure supported on an Axiom A attractor has the following two properties: (i) the empirical measure starting at a typical point near the attractor converges weakly to the SRB measure; (ii) the pushforward of any Lebesgue-absolutely continuous probability measure supported near the attractor converges weakly to the SRB measure. The first property is known as the “physical measure” property, and has been extensively studied and generalised. We will refer to the second property as the “attracting measure” property. It describes “mixing” behaviour, but in a more experimentally accessible way than just saying that the invariant measure itself is mixing: it can be expressed as a decay of “operational correlations” which make reference to the Lebesgue measure, as opposed to decay of “classical” correlations defined purely with respect to the invariant measure. There are various situations in the sciences, such as in climate science, where attractors have zero Lebesgue measure, and the question of whether such attractors support an attracting measure seems to be of high physical relevance. And yet, there appears to be very little literature addressing this question. (For example, is it known whether the physical measure on the classical Lorenz attractor is attracting?) I will present a topological generalisation of the original result of Bowen and Ruelle that establishes the attracting measure property for Axiom A flows.
Tomas Persson (Lund)
Dynamical Borel–Cantelli lemmata, shrinking targets and recurrence
This lecture will give the basics of dynamical Borel–Cantelli lemmata and shrinking targets. Some proofs will be given in at least simple cases. I will also talk about recurrence and other related things.
Mike Todd (St Andrews)
Capturing clustering in extreme values
We build new tools to handle clustering patterns of extreme values in dynamical systems. The main tool I’ll discuss is a new type of point process which fully captures clustering behaviour. I’ll illustrate this with a simple dynamical system. This is joint work with Ana Cristina Freitas and Jorge Freitas.