# ETDS 2011-12

### Welcome to the Ergodic Theory and Dynamical systems seminars web page! The seminars are held on Tuesdays at 14:00 in Room B3.02 - Mathematics Institute

Organizer: Davoud Cheraghi

Term 3

• Tuesday 29 May 2012
Alastair Fletcher (Warwick)
Decomposing diffeomorphisms of the sphere

• Tuesday 12 June 2012
Katsutoshi Shinohara (Uni)
On the (non)-minimality of free-semigroup actions on the interval C^1-close to the identity

Abstract:
We consider (attracting) free semigroup actions (with two generators) on an interval.
It is known that, if those two maps are sufficiently C^2-close to the identity, then
there is a restriction on the shape of the (forward) minimal set. Namely, it must be
an interval. (This statement is not accurate. I will give the precise statement in my
talk.) In this talk, I will explain that the similar argument fails in C^1-topology.

• Thursday 14 June 2012
Arnaud Cheritat (Institut de Mathematique de Toulouse)
The talk is moved to Friday!

• Wednesday 20 June 2012 - room MS.05
Olga Lukina (University of Leicester)
Hierarchy of graph matchbox manifolds
Abstract:
We study a class of graph foliated spaces, or graph matchbox manifolds, initially
constructed by Kenyon and Ghys. For graph foliated spaces we introduce a quantifier
of dynamical complexity which we call a level. We develop the fusion construction,
which allows us to associate to every two graph foliated spaces a third one which
contains the former two in its closure. Although the underlying idea of the fusion
is very simple, it gives us a powerful tool to study graph foliated spaces.
Using fusion, we prove that there is a hierarchy of graph foliated spaces at infinite
levels. We also construct examples of graph foliated spaces at low levels.

Term 2

• Tuesday 17 January 2012
Mark Pollicott (Warwick)
The Schottky-Klein Function and Ergodic Theory
• Tuesday 24 January 2012

Abstract: Mating is a partially defined operation which sends pairs of normalized
polynomials of the same degree to normalized rational maps of that degree.
As proposed by Douady, the recipe is to glue filled-in Julia sets back-to-back along
opposite prime ends, in the hope of obtaining a branched self-cover of a topological
sphere which is suitably conjugate to a unique normalized endomorphism of the
Riemann sphere. We will show that this procedure yields a highly discontinuous
map between the relevant parameter spaces.
• Tuesday 31 January 2012
Davoud Cheraghi (Warwick)
Optimal estimates on perturbed Fatou coordinates
• Tuesday 7 February 2012
Dimitry Turaev (Imperial College London)
On stickiness and flicker-noise
• Tuesday 14 February 2012
Peter Hazard (Warwick)
Entropy without Periodic Points in Dimension Two

Abstract: In 1980 A. Katok showed that if a $C^{1+\alpha}$-diffeomorphism $f$ of
a compact surface has positive topological entropy then it has a homoclinic point.
Consequently $f$ possesses a horseshoe, and hence infinitely many periodic orbits.
It was asked if this held for $f$ with lower regularity. M. Rees later gave a counter-
example, constructing a homeomorphism of the 2-torus with positive entropy which
is minimal.
• Tuesday 21 February 2012
James Langley (University of Nottingham)
Non-real zeros of derivatives of real meromorphic functions

Abstract: Around 1911 Wiman conjectured that if an entire function is real on the
real axis and, together with its second derivative, has only real zeros, then the
function belongs to the Laguerre-Polya class, from which it follows that all derivatives
have only real zeros. This has now been proved via a combination of results by
several authors including the speaker, as have some related conjectures advanced
by Polya in the 1940s. The talk will survey results in this direction and report on
recent progress towards an analogue of the Polya-Wiman conjectures for functions
which are meromorphic rather than entire.
• Tuesday 28 February 2012
Ian Morris (University of Surrey)
Structure of extremal matrix products

Abstract: The joint spectral radius of a finite or compact set of dxd matrices is the
maximum possible exponential growth rate of long products of matrices drawn from
that set. An influential conjecture of J. Lagarias and Y. Wang stated that for any
finite set of matrices, this optimal rate of growth is achieved by a periodic product.
We discuss counterexamples and describe the properties which must be satisfied
by sequences of extremal growth.
• Tuesday 6 March 2012
Boguslaw Zegarlinski (Imperial College London)
TBA
• Tuesday 13 March 2012
John Osborne (Open University)
Spiders' webs and locally connected Julia sets of transcendental entire functions

Abstract: In this talk, I will explore some links between the local connectedness of
the Julia set of a transcendental entire function, and a particular geometric form
of the Julia set known as a spider's web (first defined by Rippon and Stallard).

Term 1
• Tuesday 11 October 2011
Rodney Halburd (University College London)
Title: Analysis, arithmetic and integrability

Abstract: The behaviour of solutions of differential equations in the complex domain
has long been used as a detector of integrability. In this talk, complex analytic and
number theoretic properties of solutions of difference equations will be discussed
that correlate well with integrability. Classification results will be presented showing
how these properties naturally single-out the discrete Painlev\'e equations from
more general classes of equations.
• Tuesday 18 October 2011
Jörn Peter (Kiel, Germany)
Title:Hausdorff measure of Julia sets for bounded type functions of finite order
• Tuesday 25 October 2011
No meeting this week!
• Tuesday 1 November 2011
Alex Clark (University of Leicester)
Title: Dynamics of matchbox manifolds

Abstract: Matchbox manifolds are a particular type of lamination or foliated space
in which the transverse space is totally disconnected. In this talk we will review the
basic definitions and constructions and examine recent results with Steve Hurder
establishing connections between the dynamics of matchbox manifolds and their
topology. We will also discuss ongoing research with Hurder and Lukina that leads
to a general framework for studying matchbox manifolds.

• Tuesday 8 November 2011
Franco Vivaldi (Queen Mary, London)
Regular motion and anomalous tranport in a piecewise isometric system
• Tuesday 15 November 2011,
Yusuke Okuyama (Kyoto Institute of Technology)
Singularities of Schro\"oder maps and unhyperbolicity of rational functions

Abstract: One of the aims of this talk is to give a formulation of Fatou's hyperbolic
density problem for unicritical families in terms of Schro\"oder (Poincar\'e) maps
with respect to repelling periodic points of these unicritical polynomials. We really
study the relationship between transcendental singularities or asymptotic values
of Schro\"oder maps and un(semi)hyperbolicity of rational functions using both
dynamics of rational functions and covering theoretical results on meromorphic
functions.

This talk is based on our joint work with David Drasin (Purdue).

• Tuesday 22 November 2011, 2-3pm in Room B3.02
Alfonso Sorrentino (Cambridge)
Minimal average action and Hamiltonian dynamics

Abstract: In the variational study of Tonelli Hamiltonian systems an important role
is played by the so-called Mather’s minimal average action (also known as
"beta-function" or "effective Lagrangian"), with particular attention to its regularity
and symplectic properties. In this talk I'll discuss how, in the case of Hamiltonians
on closed surfaces, its differentiabilty relates to the existence of invariant
Lagrangian graphs and to the integrability of the system. Time permitting, I'll also
describe a geometrical interpretation of this function in terms of the asymptotic
distance from the identity in the group of Hamiltonian diffeomorphisms.
• Tuesday 22 November 2011, 4-5pm in Room B3.03
Vered Rom-Kedar (Weizmann Institute)
A saddle in a corner - a model of atom-diatom chemical reactions

Abstract: A geometrical model which captures the main ingredients governing
atom-diatom collinear chemical reactions is proposed. This model is neither
near-integrable nor hyperbolic, yet it is amenable to analysis using a combination
of the recently developed tools for studying systems with steep potentials and
the study of the phase space structure near a center-saddle equilibrium. The
nontrivial dependence of the reaction rates on parameters, initial conditions
and energy is thus qualitatively explained. Conditions under which the phase
space transition state theory assumptions are satisfied and conditions under
which these fail are derived. Extensions of these ideas to other impact-like
systems and to other models of reactions will be discussed.

Joint works w L. Lerman and M. Kloc.
• Tuesday 29 November 2011
Jon Aaronson (Tel-Aviv University)
On the categories of weak mixing in infinite measure spaces

Abstract: I will discuss some propositions such as "in general" an infinite measure
preserving trasformation is subsequence rationally weakly mixing, but not rationally
weakly mixing.
• Tuesday 6 December 2011
Martin Rasmussen (Imperial college London)
Morse decomposition of nonautonomous dynamical systems