# Ergodic Theory and Dynamical Systems Seminar 2010/11 - Term 2

### Term 2 2010/11 - The seminars are held on Tuesdays at 14:00 in Room B3.02 - Mathematics Institute

#### Organisers: Andrew Ferguson

• Tuesday 11 January 2011
NO SEMINAR
• Tuesday 18 January 2011
Nikita Sidorov (Manchester)
On a family of explicit counterexamples to the Finiteness Conjecture

The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth. J. C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real d × d matrices satisfies the finiteness property (the Finiteness Conjecture). However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the Finiteness Conjecture exist, but no explicit counterexample has so far been given.

The purpose of this talk is to present the first completely explicit family of 2 × 2 matrices which prove to be counterexamples to the Finiteness Conjecture. En route we will meet Sturmian sequences, continued fractions and even Liouville numbers!

This work is joint with Kevin Hare, Ian Morris and Jacques Theys.

• Tuesday 25 January 2011
Yutaka Ishii (Ecole Polytechnique)
Iterated monodromy groups for Henon maps

In this talk I relate several combinatorial descriptions for the Julia sets of hyperbolic polynomial diffeomorphisms of $C^2$-norm.: external angles by E.Bedford and J.Smillie, automata by R.Oliva and Hubbard trees by myself. Iterated monodromy groups, originally introduced for partial self-coverings of arcwise connected spaces, are defined for such polynomial diffeomorphisms and are used to construct automata from Hubbard trees.

• Tuesday 1 February 2011
One day ergodic theory meeting
• Tuesday 8 February 2011
Felipe Ramirez (Bristol)
Smooth cohomology over higher-rank abelian group actions

This talk will be about smooth real-valued cocycles over actions on homogeneous spaces. The focus will be abelian actions generated by commuting flows. It is known that the obstructions to solving the cohomology equation for an Anosov flow come from periodic orbits, and for higher-rank Anosov actions there are no obstructions. The goal of this talk is to discuss similar results for certain parabolic actions.

• Tuesday 15 February 2011
David Ruelle (IHES)
Can one differentiate SRB states with respect to dynamics?
• Tuesday 22 February 2011
Charles Walkden (Manchester)
Dynamical Weierstrass functions
• Tuesday 1 March 2011
Thomas Kempton (Warwick)
A Growth Rate For Beta Expansions

For typical real numbers x and beta, beta expansions of x are not unique. We consider some questions relating to the random beta transformations of Dajani and Kraaikamp, which allow us to simultaneously consider all beta expansions of a given number. In particular, we show that for typical x and beta, the number of words of length n that can be extended to beta expansions of x grows exponentially in n.

• Tuesday 8 March 2011
Richard Sharp (Manchester)
Fluctuation theorems and shrinking intervals
• Tuesday 15 March 2011
Georg Ostrovski (Warwick)
A Dynamical System Motivated by Games: Fictitious Play Dynamics