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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.

We will be having some online meetings to replace the activities we had planned.

NEXT ONLINE MEETING: Wednesday 24 June 2020, starting at 2:00pm (UK time).

The Zoom link for the meeting is https://us02web.zoom.us/j/87804275852

Richard

Schedule

2:00pm Simon Baker (Birmingham)

Title: Iterated function systems with super-exponentially close cylinders

Abstract: Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb R$ does not equal its expected value, then the underlying iterated function system contains an exact overlap. Significant progress on these conjectures has been made by Hochman and Shmerkin. They both proved results to the effect that if the dimension of a self-similar measure in $\mathbb R$ does not equal its expected value, then there are cylinders which are super-exponentially close at all sufficiently large scales. This naturally leads us to the following question: Do there exist iterated function systems that do not contain exact overlaps yet have cylinders which are super-exponentially close at all sufficiently large scales? In this talk I will discuss a recent paper where I show that such iterated function systems do exist.


3:00pm-3:15pm Break


3:15pm-4:15pm Liviana Palmisano (Durham)

Title: Attractors and their stability

Abstract: One of the fundamental problems in dynamics is to understand the attractor of a system, i.e. the set where most orbits spent most of the time. As soon as the existence of an attractor is determined, one would like to know if it persists in a family of systems and in which way i.e. its stability. Attractors of one dimensional systems are well understood, and their stability as well. I will discuss attractors of two dimensional systems, starting with the special case of Henon maps. In this setting very little is understood. Already to determine the existence of an attractor is a very difficult problem. I will survey the known results and discuss the new developments in the understanding of attractors, coexistence of attractors and their stability for two dimensional dynamical systems.