## PhD studentships available

There are a number of EPSRC-funded PhD studentships available within the Centre for Systems Dynamics and Control: these include the following which offer full funding to UK students and fees-only to EU students non-resident in the UK. The deadline for applications for the following is 10th January 2018:

- Cortical Network Models to Understand Differential Input Response Properties During Active and Silent States – Mathematics – EPSRC DTP funded PhD Studentship
- Critical Cascades of Fast-slow Dynamical Systems – Mathematics – EPSRC DTP funded PhD Studentship
- Understanding Extremal Processes in Dynamical Systems – Mathematics – EPSRC DTP funded PhD Studentship
- Understanding Extremes in Low Dimensional Dynamical System Weather Models – Mathematics – EPSRC DTP funded PhD Studentship

## One-day study in ergodic theory and dynamical systems

We are pleased to announce the second one-day study in Ergodic Theory and Dynamical Systems to be held at the University of Exeter on 17th November 2017. If you have any enquiries please ask .

**Programme:**

**11:30-12:30 Jimmy Tseng (Exeter University)**

Title: An introduction to homogeneous dynamics and its interactions with number theory

Room: Peter Chalk 1.1

**12:30-13:30 Pedro Peres (Exeter University)**

Title: Embeddings of Interval Exchange Transformations into Planar Piecewise Isometries

Room: Peter Chalk 1.1

**13:30- 15:30 Lunch**

**15:30-16:30 Samuel Roth (Opava University, CZ)**

Title: Constant Slope Models and Perturbation

Room: Harrison 254

**16:30-17:30 Zuzana Roth (Opava University, CZ)**

Title: Li-Yorke sensitivity and a conjecture of Akin and Kolyada

Room: Harrison 254

**Speakers and abstracts:**

**Jimmy Tseng (Exeter University)**

**An introduction to homogeneous dynamics and its interactions with number theory**

I will introduce the study of homogeneous dynamics through a particular homogeneous space, namely the space of unimodular lattices, and a particular flow, namely a suitable diagonal flow. I will show how the dynamics corresponds to Diophantine approximation, which is a subfield of number theory concerned with how well real objects (numbers, vectors, matrices) are approximated by suitable rational ones.

We will discuss ergodic properties of the diagonal flow and its consequences for number theory. In particular, I will mention a recent paper of mine, joint with Athreya and Parrish, which gives us a conceptually simple way of showing a classical theorem in Diophantine approximation.

**Pedro Peres (Exeter University)**

**Embeddings of Interval Exchange Transformations into Planar Piecewise Isometries**

Although piecewise isometries (PWIs) are higher dimensional generalizations of one dimensional interval exchange transformations (IETs), their generic dynamical properties seem to be quite different. In this talk I will consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences. I will introduce a family of PWIs with apparent abundance of invariant nonsmooth fractal curves supporting IETs. I will recall some notions used in the study of IETs, introduce some new tools and describe how they can be used to establish some results on the existence of embeddings of IETs into PWIs. (Joint work with P. Ashwin, A. Goetz and A. Rodrigues).

**Samuel Roth (Opava University, CZ)**

**Constant Slope Models and Perturbation**

We work in the space of transitive, piecewise monotone maps of a fixed modality m with the topology of uniform convergence. There is an operator on this space which assigns to a map its constant slope model. This operator is discontinuous at points (maps) where perturbation can lead to a jump in entropy. Alseda and Misiurewicz conjectured that these are the only discontinuity points. We confirm the conjecture by a technique of “counting preimages.” (joint work with Michal Malek)

**Zuzana Roth (Opava University, CZ)**

**Li-Yorke sensitivity and a conjecture of Akin and Kolyada**

Akin and Kolyada in 2003 [1] conjectured that every minimal system with a weak mixing factor, is Li-Yorke sensitive. Our interest in this problem comes from a survey paper by Li and Ye [2] which came out last year. This talk will deal with a proof of the conjecture for minimal 2-point extensions of weak mixing systems, which became an inspiration for a more complex solution by Mlichova [3]. The talk will be closed with results that showed up in the last year. (joint work with Samuel Roth)

Bibliography:

[1] Ethan Akin and Sergii Kolyada. Li-Yorke sensitivity. Nonlinearity , 16(4):1421, 2003.

[2] Jian Li and Xiangdong Ye. Recent development of chaos theory in topological dynamics. Acta Mathematica Sinica, English Series}, 32(1):83–114, 2016.

[3] Michaela Mlichova. Li-Yorke sensitive and weak mixing dynamical systems. Journal of Difference Equations and Applications, 0(0):1–8, 0.

## New staff member: Jimmy Tseng

Dr. Jimmy Tseng has joined us as a new member of academic staff. His research focuses on homogeneous dynamics, ergodic theory, number theory (especially Diophantine approximation), geometry, and the interactions of these fields. He has previously held teaching and research positions at the University of Bristol, the University of Illinois at Urbana-Champaign, and Ohio State University.

## Workshop on neural networks: from dynamical systems to psychology, 24th Nov 2017

This workshop will take place at the University of Exeter on Friday 24th November 2017

Networks are a highly topical subject for mathematical research in dynamical systems, where new challenges are being addressed, new methodologies developed and surprising connections uncovered. The workshop will discuss recent research on the mathematical and computational modelling of neural-inspired dynamic networks, as well as exploring connections to clinical-facing research.

For programme, speakers and registration, see

http://emps.exeter.ac.uk/mathematics/research/pmh/events/

## Ergodic Theory meeting in Exeter, April 2017

We are planning a number of talks on Wednesday 12th and Thursday 13th April 2017.

University of Exeter, Harrison Building, Harrison 203.

**Wednesday 12th April 2017
**

**14.00-15.00 Alexander Adam (UPMC)**

** Title: Resonances for Anosov diffeomorphisms**

Abstract: Deterministic chaotic behavior of invertible maps T is appropriately described by the existence of expanding and contracting directions of the differential of T. A special class of such maps are Anosov diffeomorphisms. Every hyperbolic matrix M with integer entries induces such a diffeomorphism on the 2-torus. For all pairs of real-analytic functions on the 2-torus, one defines a correlation function for T which captures the asymptotic independence of such a pair under the evolution T^n as $n\to\infty$.

What is the rate of convergence of the correlation as $n\to\infty$, e.g. what is its decay rate? The resonances for T are the poles of the Z-transform of the mereomorphic continued correlation function. The decay rate is well-understood if T=M. There are no non-trivial resonances of M. In this talk I consider small real-analytic perturbations T of M where at least one non-trivial resonance of T appears. This affects the decay rate of the correlation.

**15.05-16.05 Thomas Jordan (Bristol)**

** Title: Measures of maximum dimension for self-affine sets**

Abstract: (Joint work with Jonathan Fraser and Natalia Jurga) Self-affine sets are sets which made up of affine copies of themselves. The Hausdorff dimension of such sets has been a long standing research topic. For self-similar and self-conformal systems a standard approach has been to use the thermodynamic formalism to find a Gibbs measure on the associated shift space which projects to a measure which will have the same dimension as the set. By using the subadditive thermodynamic formalism the same approach can be used for self-affine sets but with much greater difficulties. It is known due to Kaenmaki that a large class of self-affine sets will have such a measure of maximal dimension (such measures are often called Kaenmaki measures). In this talk we’ll give this background before looking at the properties this measure will have in particular cases (diagonal, positive, irreducible). The new work covered will be what measure we would expect to be the measure of maximal dimension in a situation where the iterated function system is made up of diagonal and anti-diagonal matrices and how the structure of such systems can be used to show it is indeed often a measure of maximal dimension.

**16.30-17.30 Ian Melbourne (Warwick)**

** Title: Singular hyperbolic flows**

Abstract: The classical Lorenz attractor is an important example of a singular hyperbolic attractor and its statistical properties are very well understood. However, many of these results rest heavily on the fact that certain stable foliations are smooth.

In this talk, we discuss the general situation where the stable foliation need not be smooth. In addition to clarifying existing results on existence of spectral decompositions and SRB measures, we extend many of the statistical properties for the classical Lorenz attractor to general three-dimensional singular hyperbolic flows. Our results hold also in higher dimensions (for codimension two singular hyperbolic flows). This is joint work with Vitor Araujo.

**Thursday 13th April 2017**

**11.30-12.30 Tomas Persson (Lund)**

** Title: Shrinking targets in parametrised families**

Abstract: I will talk about a joint work with Magnus Aspenberg. We consider a parametrised family of piecewise expanding interval maps $T_a$ and a point $x(a)$, and study the following shrinking target problem: For which parameters $a$ is $T_a^n (x(a))$ inside a shrinking neighbourhood of a point $y$ for infinitely many $n$? We give upper and lower bounds for the set of such parameters. Our results are generalisations of several previous results for specific families. The proofs rely on techniques originating from Benedicks and Carleson, and in particular on a result by Schnellmann on typical points.

**14.00-15.00 Viviane Baladi (CNRS, IMJ-PRG, UPMC)**

** Title: Linear response for discontinuous observables**

Abstract: Linear response formulas describe how the physical measure of a dynamical system reacts to perturbations of the dynamics. For hyperbolic dynamics, linear response is usually stated for differentiable observables only. Discontinuous observables involving thresholds (Heaviside functions) appear naturally in extreme value theory. We present our recent results with Kuna and Lucarini giving sufficient conditions, on observables allowing thresholds, ensuring linear response. Our proof uses the fine properties of anisotropic Banach spaces. This will also be an opportunity to give a survey talk on anisotropic spaces suitable for transfer operators of hyperbolic dynamical systems.

**15.10-16.10 Damien Thomine (Paris Sud)**

** Title: Hitting probabilities, potential kernel and ergodic theory**

Abstract: Given a recurrent random walk, there is a simple relationship between the probability that an excursion from the origin hits a given site, and the (symmetrized) potential kernel of the random walk. The classical proof draws from harmonic analysis. We give here a new proof of this relationship, which uses tools from ergodic theory rather than from harmonic analysis. As a consequence, we are able to generalize these results to contexts for which no simple harmonic objects are available. This is joint work with Françoise Pène (Université de Brest).

More details will be update here in due course – contact Dalia Terhesiu or Peter Ashwin if you have any questions.

## EPSRC DTP Studentships in Exeter, deadline January 2017

A number of PhD studentships are funded by the Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Partnership with Exeter to commence in September 2017. The projects are funded on a competitive basis. Usually the projects that receive the best applicants will be awarded the funding. The studentships will provide funding for a stipend which, is currently £14,296 per annum for 2016-2017, research costs and UK/EU tuition fees at Research Council UK rates for 42 months (3.5 years) for full-time students, pro rata for part-time students.

The link to the list of projects is here (note that the list is long, applied maths & dynamics projects are mixed with others):

http://www.exeter.ac.uk/postgraduate/research-degrees/mathematics/funding/

## Positions available (in Exeter and elsewhere)

A 3-year position as a Postdoctoral Research Fellow is available in the research group (working with Jan Sieber:

http://www.jobs.ac.uk/job/AUU269/postdoctoral-research-fellow/

Also, internationally, several tenure track/permanent positions are open:

McGill (tenure track, info from Tony Humphries):

http://www.mathjobs.org/jobs/jobs/9249

DTU Lyngby (permanent, info from Frank Schilder):

http://www.compute.dtu.dk/om_os/ledige_stillinger/job?id=4778215b-b1f1-4a38-b927-666abf94cbb8