# Ergodic Theory & Dynamical Systems Seminar 2009-10

## Autumn Term 2009/2010

These will take place on **Tuesdays at 2:00pm in B3.02 **(unless otherwise indicated) in the **Mathematics Institute**.

For further information about Ergodic Theory and Dynamical Systems Seminars,

contact one of the organisers: **Charlene Kalle**, or **Davoud Cheraghi**. GO TO TERM 3

## Term I

- 29th September 2009

**Mike Field**(Houston)

*Exponential mixing for Hyperbolic flows*

- 13th October 2009

**Kenneth Falconer**(St. Andrews)

*Self-affine fractals and multifractals*

- 20th October 2009

**Davoud Cheraghi**(Warwick)

*Typical trajectories of quadratic polynomials with positive area Julia sets*

- 27th October 2009

**Anders Öberg**(Uppsala)

*Uniqueness and mixing rates for g-measure*

- 3rd November 2009

**Thomas Kempton**(Warwick)

Factors of Gibbs measures for subshifts of finite type

- 10th November 2009

**Charlene Kalle**(Warwick)

*β-expansions and multiple tilings*

- 17th November 2009

**Alastair Fletcher**(Warwick)

Quasiregular dynamics

- 24th November 2009

**Jean-Marc Gambaudo**(Nice - Sophia Antipolis)

On the stability of quasicrystals

- 1st December 2009

**Philip Rippon**(Open)

Boundaries of escaping Fatou components of transcendental entire functions

- 8th December 2009

**Freddie Exall**(Liverpool)

TBA

## Term II

- 12th January 2010

**Carlangelo Liverani**(Roma Tor Vergata)

*Some rigorous results in linear response theory*

This week the seminar will be in Room B3.01.

- 19th January 2010

**Richard Sharp**(Manchester)

*Length spectra of negatively curved manifolds*

- 26th January 2010

**Lasse Rempe**(Liverpool)

*Measurable dynamics of transcendental functions*

- 2nd February 2010

**Henk Bruin**(Surrey)

*On the dynamics of subtractive algorithms*

**Abstract:**There are various algorithms used to give simultaneous approximations of d

- 9th February 2010

**Gwyneth Stallard**(Open university)

*Dimensions of escaping sets of transcendental entire functions*

- 16th February 2010

**Sara Munday**(St-Andrews)

*An introduction to*-*Farey-Luroth and*-*Luroth systems and*

some strong renewal results

- 23th February 2010

**Andy Ferguson**(Warwick)

Escape rates for Gibbs measure

**Abstract:** In this talk I will present a result relating to the convergence of the spectral radius of a singularly perturbed transfer operator.

I will then discuss applications to the convergence of escape rates and topological pressure for subshifts of finite type.

This is joint work with Mark Pollicott.

- 2nd March 2010

**Edward Crane**(University of Bristol)

*The Simple Harmonic Urn*

**Abstract**: Urn models have a long history as thought experiments in statistical physics and biology, the best-known examples being

Polya's urn and the Ehrenfest urn. The behaviour of generalized Polya urns is well understood in the strictly positive case, where the

asymptotic behaviour depends on the spectrum of the reinforcement matrix. A morerecent example is the OK Corral model, a

two-colour urn model with negative eigenvalues, analysed by Kingman and Volkov by coupling with a pair of independent death

processes. In this talk we present a two-colour urn model with complex eigenvalues, which we call the simple harmonic urn.

We analyse it with a similar coupling method, revealing a connection with the renewal process with uniform interarrival times.

In passing we prove new sharp results about the uniform renewal process, essentially giving a sharp estimate of exponential decay

of correlations for a system induced from a shift space. Exact expressions for the transition probabilities involve the Eulerian numbers,

which count permutations with a specified number of descents. We show that the simple harmonic urn is transient, but only barely

- after a slight modification it becomes recurrent. This is neatly illustrated by a percolation model whose directed paths provide a

coupling of many instances of the simple harmonicurn and whose planar dual encodes the modification. Time permitting, we will

discuss two embeddings of the simple harmonic urn model in stationary processes and describe some open questions.

- 9th March 2010

**Toby Hall**(Liverpool.)

*Paper surfaces and dynamical limits*

**Abstract:** A paper surface is one which is obtained by making identifications along the sides of polygons in the plane: the difference

from the usual construction of surfaces is that the identifications can becarried out along infinitely many arcs in the boundaries of the

polygons. I will discuss the topological, metric, and complex structures of paper surfaces, and explain their relevance in taking limits

of sequences of two-dimensional dynamical systems. This is joint work with Andre de Carvalho (Sao Paulo).

- 16th March 2010

**Hirojuki Inou**(Kyoto)

*Similarities in parameter spaces*

**Abstract**: It is well-known that (the boundary of) the Mandelbrot set is self-similar. By computer pictures of families of cubic

polynomials, you can easily see much more complicated structures. For example, quadratic Julia sets and the Mandelbrot

set are homeomorphically (or even quasiconformally) embedded into some one-parameter families of cubic polynomials.

I will summarize known results on such similarities and related results. I will also present some works in progress and conjectures.

## Term III

- 28th April 2010

**Renaud Leplaideur**(Université de Brest)

*Selection of Maximizing Measure at Temperature Zero in the Shift*

This week the seminar will be on**Wednesday**at 2pm in Room MS.04. - 11th May 2010

**Matthew Nicol**(University of Houston)

*A Borel-Cantelli Lemma For Non-uniformly Expanding Dynamical Systems* - 18th May 2010

**Jaime Sanchez**(Warwick)

*Shape theory in the realm of dynamical systems* - 25th May 2010

**Corinna Ulcigrai**(University of Bristol)

*Interval exchanges techniques to prove mixing properties of area preserving flows*

- 1st June 2010

**Name**(Univ)

Title

- 8th June 2010

**Mark Pollicott**(Warwick)

Dynamical zeta functions: a*miscellany* - 15th June 2010

**Yongcheng Yin**(Fudan University (China))

*On bounded Fatou components for polynomials*

**Abstract:** In this talk, we consider the local connectivity of the boundary of bounded Fatou components and Julia sets for polynomials.

We prove that (1) if U is an attracting or a parabolic bounded Fatou component, then its boundary is locally connected;

(2) if z is not prperiodic and its omega limit set intersects with the boundary of an attracting or a parabolic bounded Fatou component,

then the Julia set is locally connected at z.

This is a joint work with Pascale Roesch.

- 22nd June 2010

**Andras Mathe**(Warwick)

*Geometric properties of self-similar sets and measures*

**Abstract:**Let K be a self-similar set in a Euclidean space (satisfying the strong separation condition). I will study the

intersection of K with its similar copy g(K). It turns out that this intersection has positive measure if and only if its relative

interior in K is non-empty. Here "measure" is either the Hausdorff measure or any self-similar measure of K. I will also

give necessary algebraic conditions for those maps g for which the intersection of K and g(K) is of positive measure.

Most of the results generalize to self-conformal sets (attractors of IFS of smooth conformal maps), endowed with a

Gibbs measure. I will also mention some open problems. (Based on joint work with M. Elekes and T. Keleti.)

- 29th June 2010

**Tom Sharland**(Warwick)

*Clustering and the mating of quadratic polynomials*

**Abstract**: Let F be a quadratic rational map on the Riemann sphere. Suppose that a subcollection of the critical orbit Fatou

components meet at a periodic point c. We then call c a cluster point for the map F. I will discuss how maps with cluster

points arise naturally as the result of matings of polynomials and what we can say about the properties of the polynomials

that create clustering under mating, including some results using the language of symbolic dynamics of quadratic polynomials.

I will also describe how, for periods one and two, a very simple set of combinatorial data completely defines a rational map

up to a M\"obius transformation. If there is time, I will discuss how these results generalise in the case that F is a bicritical

rational map of degree d and the difficulties one encounters when dealing with cluster cycles of periods 3 or greater.