# Ergodic Theory and Dynamical Systems Seminar 2010/11 - Term 1

### Term 1 2010/11 - The seminars are held on Tuesdays at 14:00 in Room B3.02 - Mathematics Institute

#### Organisers: Andrew Ferguson, Mark Pollicott

• Tuesday 5 October 2010
Dalia Terhesiu (Surrey)
Operator renewal theory for infinite measure preserving systems
• Tuesday 12 October 2010
Averaging, passages through resonances, and captures into resonance in dynamics of charged particles
• Tuesday 19 October 2010
Wael Bahsoun (Loughborough)
Invariant densities and escape rates: rigorous and computable approximations in the $L^{\infty}$-norm.
• Tuesday 26 October 2010
Pablo Guarino (IMPA)
$C^1$ rigidity for smooth critical circle maps

It has been proved by Khanin and Teplinsky that two analytic critical circle maps with the same irrational rotation number (and the same odd degree of the critical point) are $C^1$ conjugate to each other. We extend this rigidity result to $C{^\infty}$ critical circle maps. This is a joint work with my advisor Welington de Melo.

• Tuesday 2 November 2010
Phil Rippon (Open University)
Slow escaping points of entire functions

• Thursday 4 November 2010 - Room MS.04 at 15:00
Anders Öberg (Uppsala)
Bernoulli g-measures via joining

• Tuesday 9 November 2010
Trevor Clarke (Warwick)
Regular or stochastic dynamics in families of unimodal maps

About fifteen years ago, Palis conjectured that typical dynamical systems should possess good statistical properties. Through the work of Avila, Lyubich, de Melo and Moreira, this has been proved for unimodal maps with a non-degenerate critical point. I will explain the differences in the higher degree case, and show how to remove the condition on the critical point.

• Tuesday 16 November 2010
No seminar
TBA

• Tuesday 23 November 2010
Anish Ghosh (UEA)
Rational points on spheres

I will explain what the ergodic theory of group actions tells us about the distribution of points of arithmetic interest on spheres.

• Tuesday 23 November 2010 - Room B1.01 at 16:00
Omri Sarig (Weizmann Institute and Penn State)
Pseudo orbits for non uniformally hyperbolic surface diffeomorphisms

A "pseudo orbit" for a dynamical system $f$ is a sequence of points $(x_i)_i$ s.t. $dist(f(x_i),x_{i+1})<\epsilon$ for all $i$. If $f$ is "uniformly hyperbolic" (e.g. the "cat map") and epsilon is small enough, then every pseudo orbit is close to a real orbit (Anosov). Anosov's theorem allows one to construct orbits from "nearest neighbour" constraints. I will discuss a similar tool for producing orbits from nearest neighbour constraints, which works for $C^r$ ($r>1$) surface diffeomorphism with positive topoogical entropy.

• Tuesday 30 November 2010
Roland Zweimueller (Surrey)
Globally coupled ergodic systems with bistable thermodynamic limit;

I will report on joint work with Gerhard Keller (Erlangen) and Jean-Baptiste Bardet (Rouen). We study systems consisting of a large number of identical ergodic transformations which interact via a mean-field coupling rule, meaning that the evolution of each component depends on the global average of all states. Specifically, we investigate the asymptotic behaviour of the distribution of a fixed individual state in a particular model. In the "thermodynamic limit" (number of components to infinity) the evolution of these distributions is given by a nonlinear self-consistent Perron-Frobenius operator (SCPFO). For the range of coupling (strength) parameters we consider, our finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. However, within this range of parameters, the SCPFO undergoes a bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium. This gives the first rigorous explanation of a numerically observed phenomenon sometimes called the violation of the law of large numbers in mean-field coupled maps.

• Tuesday 7 December 2010
Alexander Gorodnik (Bristol)
Mixing properties of commuting automorphisms

We discuss mixing properties of commuting automorphisms of nilmanifolds. This provides an interesting example of partially hyperbolic systems that exhibit rich chaotic properties. It turns out that these properties have a number-theoretic origin. This is a joint work with Spatzier.

• Thursday 9 December 2010 - Room MS.03 at 16:00
Benito Pires (Universidade de Sao Paulo)
On $C^r$ -closing for flows on orientable and non-orientable two-manifolds

We say that a non-trivial recurrent point $p$ of a $C^r$ vector field $X$ is $C^r$-closable if there exists a one-parameter family of arbitrarily small $C^r$ perturbations of $X$ having a periodic trajectory passing through $p$ . We provide a condition under which $p$ is $C^r$-closable via twist perturbations. This is a joint work with Carlos Gutierrez.