Topology and Dynamics Seminar

Spring 2020

An introduction to the transfer operator method

Andrew Mitchell (University of Birmingham)
Friday 06 March 2020, 15:00
Lecture Theater C, Watson Building

In this series of seminars I will present an introduction to the transfer operator method. The transfer operator encodes information about an iterated map and is used to study the behaviour of dynamical systems, with applications to, for example, the calculation of Lyapunov exponents and decay of correlation. In this first talk I will present the definition and key properties of the transfer operator, and calculate the transfer operator for some well-known dynamical systems.

Growth along geodesic rays in hyperbolic groups

Stephen Cantrell (University of Warwick)
Friday 28 February 2020, 15:00
Lecture Theater A, Watson Building

Let G be a non-elementary hyperbolic group equipped with a finite generating set S. Suppose that G acts cocompactly by isometries on a space X. If we fix an origin for X then we can ask the following general question: by how much does a group element g in G displace the origin and, how does this displacement compare to the word length of g (with respect to S)? In this talk we will discuss one way of answering this question. More specifically we will study how the displacement of the origin grows as we travel along infinite geodesic rays in the Cayley graph of G.

Equicontinuity, transitivity and sensitivity

Joel Mitchell (University of Birmingham)
Thursday 06 February 2020, 14:00
Lecture Theater C, Watson Building

Robert Devaney defined chaos as a sensitive, transitive map where the set of periodic points is dense in the phase space. With an elegant proof, Banks et al showed that the the latter two properties entail the first. Since then, various analogues and generalisations of this result have been offered. Central to these theorems lie the notions of transitivity, equicontinuity, minimality and sensitivity.

In this talk I take a topological approach to dynamics and discuss sensitivity, topological equicontinuity and even continuity. I will provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but no equicontinuity point. Time permitting, I will define what it means for a system to be eventually sensitive and give a dichotomy for transitive dynamical systems in relation to eventual sensitivity.

This talk is based upon joint work with Chris Good and Robert Leek. (See `Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited’ (2020) DCDS -A 40(4).)

New approaches to overlapping iterated function systems

Simon Baker (Birmingham)
Wednesday 29 January 2020, 13:00
Lecture Theater A, Watson Building

A standard technique for generating fractal sets is to use an object called an iterated function system (or IFS for short). When an IFS satisfies some separation condition then much is known about the corresponding fractal. The situation is far more complicated when the IFS fails to satisfy any separation condition, and the pieces of the fractal overlap significantly. In this talk I will discuss a new approach for describing how an IFS overlaps. This approach uses ideas from Diophantine approximation.

Random substitutions and topological mixing

Dan Rust (Bielefeld)
Thursday 23 January 2020, 10:00
R17/18 Watson building

I’ll introduce a new class of symbolic dynamical systemassociated with ‘random substitutions’. They’re the positive-entropycousins of substitution subshifts and so many of the techniques that areused to study substitutions can analogously be utilised in the randomsetting. I’ll explain how we are able to determine when one of thesesubshifts is topologically mixing by studying an appropriateabelianisation of the substitution and which cases are still unresolved.

Joint work with Eden Provido, Lorenzo Sadun and Gwen Tadeo