# Ergodic Theory Network

This is the webpage for a network of collaborative meetings in ergodic theory, supported by a London Mathematical Society Scheme 3 grant. The universities in the network are Birmingham, Bristol, Exeter, Loughborough, Manchester, Queen Mary, St Andrews, Surrey and Warwick.

Next meeting: **Tuesday 25th June 2019** at Birmingham University, Watson Building, Lecture Theatre C

1:40pm: Christian Bick (Exeter)

Title: * Heteroclinic cycles and networks between relative equilibria on the torus*

Abstract: Vector fields on the torus which are equivariant with respect to a continuous group action arise in the study of networks of coupled oscillators. We will consider a class of such vector fields and give some recent results on the existence of heteroclinic cycles and networks between equilibria relative to the group action. Moreover, we study the asymptotic and nonasymptotic stability properties of the cycles in terms of parameters that determine the vector fields. These results are not just of interest from a theoretical point of view, but also from an applied point of view since they give explicit results on how network structure shapes the global dynamics. This is joint work with A. Lohse (Hamburg).

2:40pm: **Alex Clark** (QMUL)

Title: *Approximating spaces and dynamical systems with simpler spaces and systems*

Abstract: In this talk we will give an overview of the use of approximations of complicated spaces by simple spaces to construct inverse limit representations of spaces. In some cases these representations can also encode information about an underlying dynamical system. We will show how these representations can be used to generate new invariants that allow one to better understand and classify the original spaces and dynamical systems.

4:05pm **Charlene Kalle** (Leiden)

Title: *Frequency of zero in signed binary expansions*

Abstract: In this talk we consider a one-parameter family of dynamical systems that generate binary expansion of numbers using digits -1, 0 and 1. By the Birkhoff Ergodic Theorem the frequency of the digit 0 in typical expansions can be determined from an invariant measure that is equivalent to the Lebesgue measure. We use a surprising relation between this family of maps and Nakada's continued fraction maps to obtain an explicit expression for the density of such a measure. Then we identify the region in the parameter space where the frequency of the digit 0 in typical signed binary expansions is as large as possible.

5:05pm **Piotr Oprocha** (AGH University of Science and Technology)

Title: *On shadowing property and local dynamics*

Abstract: In this talk we will discuss consequences of shadowing property for dynamics on compact metric spaces. Among other things, we will discuss such properties as the structure of minimal sets, topological entropy or properties invariant measures. On various spaces shadowing is generic, however there is no transitivity neither any kind of expansiveness in these systems. Shadowing property allows us to better understand local aspects of dynamics in typical systems.

See here for the meeting webpage.

Future meetings:

The location and timing of meetings in the 2019-20 academic year is still to be decided.