When a game is played over time, the players can change their actions throughout the game. The choice of action can follow different learning mechanisms leading to different types of dynamics.    In this talk I shall look at the relation between the time-average of replicator dynamics (RD) and best-response dynamics (BRD), and its corresponding fictitious play (FP).  It is known that the time-average of RD converges to an invariant set under BRD but not whether given an invariant set BRD there always exists a corresponding RD orbit.

We will discuss 3-dimensional parabolic flows which are renormalised by circle extensions of Anosov diffeormorphisms (this includes nilflows on the Heisenberg nilmanifold). We use the spectral information of the transfer operators associated to these partially hyperbolic maps to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow. (Joint work with Lucia Simonelli.)

This talk starts with a generalization of the formula of Brin and Katok concerning the local metric entropy. The new formula provides a simple proof of the fact that invariant weak-Gibbs probability measures are equilibrium states. It also clarifies the role of the topological pressure on the standard definition of Gibbs property. This is a joint work with Sebastián Pérez.

The investigation of the properties of bounded linear maps on certain vector spaces (Hilbert or Banach spaces, for example) is a very rich and active area. In particular, the existence of dense orbits (that in this context is known as hyperciclicity) attracts a lot of attention, as well as the extension of classical results to this setting, like hyperbolicity and shadowing, among many others.

In this talk we discuss the dynamics of interval piecewise $\lambda$-affine contractions $f_\lambda$ where $\lambda\in(0,1)$. The map $f_\lambda$ is called asymptotically periodic if the $\omega$-limit set of every point is a periodic orbit and there are at most a finite number of periodic orbits. Under a generic assumption on $f_\lambda$, we show that the set of $\lambda\in(0,1)$ such that $f_\lambda$ is not asymptotically periodic has zero Hausdorff dimension.

Coupled populations of identical phase oscillators may give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. We consider an example of four coupled phase oscillator populations consisting of two oscillators each, such that there are two heteroclinic cycles forming a heteroclinic network. While such networks cannot be asymptotically stable, their local attraction properties can be quantified by stability indices.

I will give a survey of my recent results in this area: 1) with Jozef Bobok we have studied the dynamical properties of a typical continuous map of [0,1] which preserve the Lebesgue measure, 2) with Eleonora Catisigeras we have studied the structure of invariant measures of a typical continuous map of closed compact manifold (with or without boundary), and also for a typical homeomorphism.

We present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional and one 2-dimensional separatrices between two hyperbolic saddles-foci with different Morse indices.

This is the Dynamical Systems Group closing activity of 2019. It consists of three
40min talks, described below:

Ana Rodrigues (Exeter): Recent advances on fair measures and fair entropy

We study a topological entropy of a finitely generated semigroup G of continuous selfmaps defined on a compact metric space X. The topological entropy coincides with the limit of upper capacities of dynamically defined Caratheodory structures on X depending on G.