Dynamical Systems and Ergodic Theory

Monday 23rd June

15:10 – 15:50 Cecilia Gonzalez-Tokman (Queensland)
15:50 – 16:30 Kasun Fernando (Brunel)
16:30 – 17:10 Jens Rademacher (Hamburg)

Tuesday 24th June

15:40 – 16:20 Jamie Walton (Nottingham)
16:20 – 17:00 Andrew Mitchell (Open University)
17:00 – 17:40 Nataliya Balabanova (Birmingham)

Wednesday 25th June

15:40 – 16:00 Demetris Hadjiloucas (European University Cyprus)
16:00 – 16:20 Alejandro Rodriguez Sponheimer (Lund)
16:20 – 16:40 Jiayao Shao (Warwick)
16:40 – 17:00 Andrew Burbanks (Portsmouth)
17:00 – 17:20 Saeed Shaabanian (St Andrew’s)
17:20 – 17:40 Yiwei Zhang (Anhui)

Titles and abstracts

Natalia Balabanova – Adaptive dynamics for Prisoner’s Dilemma
Adaptive dynamics is a method of determining the best way to alter the strategy of mutants in the population in order to maximise the profit. This mathematical apparatus can be applied in game theory to the repeated donation game, otherwise known as the prisoner’s dilemma. In this talk, we are going to cover the construction of the payoff function for the prisoner’s dilemma, its symmetries, the corresponding adaptive dynamics and some of their properties.

Andrew Burbanks – Existence of periodic orbits of renormalisation operators for coupled systems
Universal scaling behaviour is observed in the bifurcation structure of families of maps. The best known example concerns period-doubling cascades in which asymptotic scaling relationships occur in parameter space, state space, and time, as the system approaches the accumulation of period-doublings. This can be explained by considering renormalisation operators that encode the scaling. The existence of periodic points of these operators, and the properties of the associated tangent maps, help to explain quantitative universality of the dynamics. We present work on the existence of renormalisation fixed points and cycles for period doubling in coupled pairs of maps. Our work involves obtaining rigorous computer-assisted bounds on operations in Banach spaces of pairs of functions. Example applications concern the FQ-type fixed point and the C-type period-two cycle, that help to explain universality of scaling in qualitatively different classes of coupled dynamical systems.

Kasun Fernando – A Dolgopyat-type estimate for a class of partially hyperbolic systems
In this talk, I will discuss a Dolgopyat-type estimate for a class of partially hyperbolic systems on the 2-Torus, and its consequences: the CLT, its higher order corrections, and exponential decay of correlations for a class of three-dimensional skew products. This is a joint work with Roberto Castorrini.

Cecilia Gonzalez-Tokman – Universal gap growth for Lyapunov exponents of perturbed matrix products
In this talk, we address the question of quantitative simplicity of the Lyapunov spectrum of bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we show explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we establish existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices, making no stationarity assumptions. Hence, our results apply to random and sequential dynamical systems alike. Joint work with Jason Atnip, Gary Froyland and Anthony Quas.

Demetris Hadjiloucas – False bubbling in Invariant Graphs of Quasiperiodically Forced Maps
The invariant subspace of a class of quasiperiodically forced maps is studied as a parameter is varied. It is shown that the invariant subspace is a unique non-trivial smooth curve which is attracting and is the graph of a smooth function. However, as the parameter is varied, a phenomenon we call false bubbling is observed. While simple iteration of the dynamics as well as Newton’s method fail to converge to the true invariant curve, an algorithm based on Singular Value Decomposition is devised and shown to converge to the true invariant curve. This is then used to make a conjecture on the existence and regularity of this invariant subspace as it loses stability beyond some critical value of the parameter. This is joint work with Paul Glendinning.

Andrew Mitchell – Substitutions and their generalisations
Dynamical systems arising from substitutions are the prototypical examples of aperiodic minimal systems and provide theoretical models for quasicrystals. In recent years, increased attention has been paid to generalisations of substitutions, including S-adic systems, staggered substitutions and random substitutions, for which the associated dynamical system can exhibit diverse properties not possible for classical substitution systems. In this talk, I will provide an overview of recent developments in the theory of substitutions and their generalisations, and discuss several pertinent open questions.

Jens Rademacher – Interaction and intrinsic dynamics of localised patterns: kinks and fronts in excitable media
The dynamics of spatially localised patterns is a characteristic phenomenon of nonlinear spatially extended systems. In the past decades many mathematical results have been obtained concerning the dynamics caused by the interaction of multiple localised patterns through spatial coupling. Much less attention has been paid to the possibility of intrinsic dynamics of a localised state. This talk discusses some aspects of both for excitable media, where the localised states are kinks or fronts. In the first part we consider a class of cellular automata, which do not feature long range spatial interaction and then turn to so-called weak interaction of related kinks in neural models. In the second part we discuss motion of single fronts by intrinsic dynamics, caused by scale-interaction between components, which admits provably chaotic dynamics. The first part is joint work with Marc Kesseböhmer (Bremen) and Dennis Ulbrich (Münster), the second with Martina Chirilus-Bruckner (Leiden) and Peter van Heijster (Wageningen)

Alejandro Rodriguez Sponheimer – Strong Borel-Cantelli Lemmas for Recurrence
The study of recurrence in dynamical systems can be dated back to the famous Poincar\’e Recurrence Theorem. One formulation of it is that, under some mild assumptions on the measure-preserving dynamical system $(X,T,\mu,d)$ , almost every point is recurrent. In other words, $\liminf_{n \to \infty}d(T^{n}x,x) = 0$ for $\mu$ -a.e. $x$ . A shortcoming of this result is that we do not obtain any quantitative information about the recurrence. For example, what can be said about the rate of convergence? What about the rate of recurrence? In 1993, Boshernitzan proved that, under some additional mild assumptions on $(X,T, mu,d)$ , one can obtain a rate of convergence for the limit $\liminf_{n \to \infty}d(T^{n}x,x) = 0$ . Since then, there have been various improvements to Boshernitzan’s result. In this talk, I will present some recent improvements in the form of dynamical strong Borel-Cantelli lemmas, giving conditions for which the limit $\lim_{n \to \infty}\frac{\sum_{k=1}^{n}1_{B(x,r_k(x))}(T^{k}x) }{\sum_{k=1}^{n} \mu(B(x,r_k(x))}=1$ holds. Such statements can be viewed as recurrence versions of dynamical strong Borel–Cantelli lemmas for the shrinking targets problem. This talk is based on joint work with Tomas Persson.

Saeed Shaabanian – A natural connection between recurrence Laws in $\phi$ -mixing dynamical systems
Recurrence in dynamical systems can be analyzed via two key laws: hitting time statistics, which study the return times to shrinking neighborhoods of a point, and escape rates, which measure how quickly orbits leave through a fixed subset of the space, interpreted as a hole. A foundational result by Bruin, Demers, and Todd (2018) established a general formula connecting these two perspectives, showing that results for one law can be transferred to the other by tuning certain parameters. This formula was originally derived in the context of interval maps. In this talk, we discuss how to extend this framework to symbolic dynamical systems with a $\phi$ -mixing measure, with one of the key applications being the generalization of the results to other dynamical systems, such as hitting of Anosov diffeomorphisms to certain subspaces, or interval maps to Cantor sets.

Jiayao Shao – Stochastic Dynamical System Methods Applied to Ship Capsize Problem
Large Deviation Theory is widely applied in the field of rare events. For example, on top of some deterministic dynamical systems, we add small noise to disturb the system to allow transition from a stable sink to a saddle. In the limitation of small enough noise, this transition would be a rare stochastic event. Under this setting, we proposed a method to investigate the probability of such an event. If it happens, the most likely trajectory of transition could be deduced. Moreover, we isolate the two fixed points by ‘Method of Division’ which largely saves computation cost and allows the case of infinite time scale. The above method will be illustrated by an engineering problem-the ship capsize problem.

Jamie Walton – Hierarchy and cut and project sets
I will introduce the two main methods of construction of aperiodically ordered patterns: by substitution (or ‘inflate, subdivide’) rules, creating patterns which may be thought of as ‘upside-down fractals’, and the cut and project method, wherein one takes an irrational slice of a periodic lattice. Many well-known aperiodic patterns, such as the Penrose tilings or recently discovered hat/spectre families of tilings (at least with particular parameters for edge-lengths), can be constructed through both procedures. I will introduce a generalised notion of a ‘pattern’ and its associated dynamical system, which gives a unified framework for studying tilings, point sets and more in the translationally FLC (finite local complexity) case. We also define when such a pattern is ‘substitutional’ in a general sense. I will explain recent results (in preparation) which establish an equivalence between unique composition of the hierarchy of such patterns and aperiodicity of their associated dynamical system, mildly generalising a well-known result of Solomyak (still only in the FLC case, but extending here to non-minimal patterns). I will then explain recent results, joint with Harriss and Koivusalo, on when a cut and project set with Euclidean total space is substitutional. We obtain a particularly simple and checkable criterion in the case that the window is assumed to be polytopal.

Yiwei Zhang – Recent progresses on typically periodic optimization problems in ergodic optimization
I will report some recent progresses on typically periodic optimization problems in ergodic optimziation theory. This is a joint work with Wen Huang, Oliver Jenkinson, Leiye Xu.