J Ize conjectured that for any absolutely irreducible representation of a compact Lie group G on a finite dimensional real vectorspace there exists an isotropy subgroup which has an odd dimensional fixed point space. If it were true it had immediate consequences in equivariant bifurcation. Lauterbach & Matthews showed that this is not the case. Their findings of three infinite families of finite groups were supplemented by extensive computer analysis showing a very difficult zoo of groups acting on R4. In this talk we will give a complete list of counter examples in R4.

The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2]. 

The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2].

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015).

This talk is about describing dynamics on primitive network objects and finding conditions that allow a good "reductive" description of network dynamics. We will give a number of examples when feedback is added.  The examples range from surprising synchrony, an example of the "bullwhip" effect and a remarkable layered network mixing synchrony and chaotic dynamics. Some of this work is part of a joint project with Ana Dias and Manuela Aguiar.

We examine the relation between a heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. In particular, we show that there are robust realizations of a large class of transitive directed graphs that are not complete (i.e. not all unstable manifolds of nodes are included) but almost complete (i.e. complete up to a set of zero measure in the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension).

Although one could naively expect that random Lorentz gases are easier to investigate than deterministic periodic ones, this seems not to be the case as essentially no results are available in the non periodic case. In this talk, I will present some general ideas towards studying random Lorentz gases and I will show how to apply them for a class of deterministic walks in random environments wit hone-dimensional uniformly expanding local dynamics. This is a joint work with Carlangelo Liverani.