One Day Ergodic Theory Meeting
There was a One Day Ergodic Theory meeting at the University of St Andrews on Thursday 14th June 2018. This is part of a network of collaborative meetings involving Bristol, Exeter, Leicester, Loughborough, Manchester, Queen Mary, St Andrews, Surrey and Warwick, funded by a Scheme 3 grant from the LMS. This followed the 2018 BMC which was held in St Andrews from 11th-14th June. The talks were held in Lecture Theatre C in the Mathematical Institute and the schedule was as follows:
1.30pm - 2.30pm: Rhiannon Dougall (University of Nantes). Critical exponents for normal subgroups via a twisted Bowen-Margulis current.
Abstract: For a discrete group Γ of isometries of a negatively curved space
X, the critical exponent δ(Γ) measures the exponential
growth rate of the orbit of a point. It is known for a certain class of
Γ0 and X, that for any normal subgroup Γ of Γ0
we have δ(Γ)=δ(Γ0) if and only if the quotient
Γ0/Γ is amenable. We will motivate this problem, and
discuss what is new: the construction of a twisted Bowen-Margulis
current on the double-boundary, which highlights a feature of
ergodicity, and extends the class for which the result is known. This is
joint work with R. Coulon, B. Schapira and S. Tapie.
2.30pm - 3.30pm: Thomas Jordan (University of Bristol). Multifractal analysis for Lyapunov exponents of self-affine iterated function systems.
Abstract: Joint work with Balazs Barany, Antti Kaenmaki and Michal Rams. For a large class of self-affine systems in ℝ2 (strongly irreducible with a reasonable separation condition) we consider level sets where the local upper and lower Lyapunov exponents take prescribed values a and b. We will find the Hausdorff dimension of these sets using a pressure function and a conditional variational principle and show the dimension varies continuously with a and b. The main technique used is to approximate a suitable pressure function on these systems via dominated systems where Lyapunov exponents correspond to Birkhoff averages of Holder continuous functions. We also need to use a recent result on the dimension of Bernoulli measures on such systems due to Barany, Hochman and Rapaport.
3.30pm - 4.00pm: Coffee break.
4.00pm - 5.00pm: Mark Demers (Fairfield University). A measure of maximal entropy for the finite horizon periodic Lorentz gas.
Abstract: While the existence and properties of the SRB measure for the billiard map associated with a periodic Lorentz gas are well understood, there are few results regarding other types of measures for dispersing billiards. We begin by proposing a naive definition of topological entropy for the billiard map, and show that it is equivalent to several classical definitions. We then prove a variational principle for the topological entropy and proceed to construct a measure which achieves the maximum. This measure is K-mixing and positive on open sets. An essential ingredient is a proof of the absolute continuity of the unstable foliation with respect to the measure of maximal entropy. This is joint work with Viviane Baladi.
For further information, please contact Jonathan Fraser or Mike Todd.