## 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

## Workshop on fast-slow dynamics

As part of the EPSRC Center for Predictive Modelling in Healthcare there will be a half-day workshop on the dynamics of fast-slow dynamical systems. Details are posted on http://emps.exeter.ac.uk/mathematics/news/event/?semID=1828&dateID=4318

All welcome!

## Upcoming workshop, 20-22nd April 2016 at Exeter

There will be a “Workshop on synchronization and oscillators with generalized coupling” hosted by the Centre on 20th to 22nd April 2016. For more information, see the workshop web pages.

## Positions available in Centre for Predictive Modelling in Healthcare

The EPSRC Center for Predictive Modelling in Healthcare, which incorporates several members from the Centre for Systems, Dynamics and Control, is looking to appoint a number of research fellows and a scientific programmer. The deadline for this round of applications is 1st March 2016.

**Research Fellow in Mathematical Modelling of Excitable Dynamics**

Based around the theme “construction and analysis of excitable dynamics in continuum systems”, the post holder will initially focus on the development of numerical and analytical reduction methods for approximation of dynamical phenomena in continuum models of excitable systems, working with Professors Vadim Biktashev and Peter Ashwin.

**Research Fellow in Mathematical Modelling of Clinical Electrophysiological Data**

Based around the theme “mathematically modelling clinically relevant electrophysiological data”, the post holder will initially focus on the development and analysis of mathematical models that can explain the mechanisms underlying electrophysiological data acquired in clinical practice, such as EEG or ECG, working with Dr Marc Goodfellow and Professor Vadim Biktashev.

**Research Fellow in Neuroendocrine Modelling and Analysis**

Based around the theme “neuroendocrine systems modelling and analysis” the post holder will work on the development of multi-scale network models of endocrine axes and hormonal secretion and understanding the amplitude and timing of exogenous delivery of hormones, working with Professors Krasimira Tsaneva-Atanasova and John Terry.

**Experimental Officer – Scientific Programmer**

The post holder will translate research quality codes into working prototypes suitable for use by end-users in healthcare (e.g. GPs, Clinicians and industry). In this regard, there is the need to communicate with both technical researchers and non-technical end-users, and to develop back-end HPC codes, as well as user-centric front-end GUIs.

## New member of staff: Dalia Terhesiu

We welcome a new member of academic staff, Dr Dalia Terhesiu, who will be joining us from Vienna in January 2016. Currently there is a fully funded (for UK students) PhD studentship available to work at the Centre for Systems, Dynamics and Control with Dr Terhesiu in the area of:

**Statistical properties of dynamical systems with infinite measure.
**

Many such systems characterized by regular variation satisfy appropriate versions of the central limit theorem. The aim of this project is to study analogous results in the absence of regular variation.

The deadline for applications is 4th January 2016, and more details are available from: http://www.exeter.ac.uk/studying/funding/award/?id=2035