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.

A well-known result of Bowen says that the periodic orbits of a hyperbolic flow become, on average, equidistributed with respect to the measure of maximal entropy as the periods tend to infinity. By introducing weightings, Parry showed that other Gibbs states could arise as limits in this way. We will discuss what happens if we restrict to Anosov flows are only consider periodic orbits which are trivial in homology.

In this talk I will discuss how to compute the entropy following backwards trajectories in a way that at each step every preimage can be chosen with equal probability introducing fair measure and fair entropy (joint with M. Misiurewicz). I will discuss some advances on the study of fair entropy for non-invertible interval maps under the framework of thermodynamic formalism, showing that the fair measure is usually an equilibrium state (joint with Y. Zhang). I will then talk about some recent results (joint with S. and Z.