One Day Ergodic Theory Meeting

When: Thursday, January 18, 2024, 2:00 PM - 5:45 PM
Where: Mathematical Sciences Building, Room MB-503 , Mile End

One Day Ergodic Theory Meeting is a "wandering" seminar series supported by an LMS grant and organised by 10 UK universities: Birmingham, Bristol, Durham, Exeter, Loughborough, Manchester, The Open University, Queen Mary, St. Andrews and Warwick. 

The January 2024 Ergodic Theory Meeting will be hosted by Queen Mary's School of Mathematical Sciences. 

14:00 - 15:00 - Çağrı Sert (Warwick)

Title: Stationary measures for SL2(ℝ)-actions on homogeneous bundles
over flag varieties

Abstract: Let Xk,d denote the space of rank-k lattices in ℝ d. Topological and statistical properties of the dynamics of discrete subgroups of G = SLd(ℝ) on Xd,d were described in the seminal works of Benoist-Quint. A key step/result in this study is the classification of stationary measures on Xd;d. Later, Sargent-Shapira initiated the study of dynamics on the spaces Xk,d. When k 6≠d, the space Xk,d is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure μ Zariski-dense in a copy of SL2(ℝ) in G, we give a classification of μ-stationary measures on Xk,d and prove corresponding equidistribution results. In contrast to the results of Benoist-Quint, the type of stationary measures that μ admits depends strongly on the position of SL2(ℝ) relative to parabolic subgroups of G. I will start by reviewing preceding major works and ideas. Joint work with Alexander Gorodnik and Jialun Li.

15:00 - 15:30 Refreshments 

15:30 - 16:30 - Natalia Jurga (St Andrew's)

Title: Hausdorff dimension of the Rauzy gasket
 
Abstract: The Rauzy gasket, which is the attractor of an iterated function system on the projective plane, is an important subset of parameter space in numerous dynamical and topological problems. Arnoux conjectured that the Hausdorff dimension of the Rauzy gasket is strictly less than 2, and since then there has been considerable interest in computing its Hausdorff dimension. After introducing the set and discussing its history, we will show how recent developments in the theory of self-affine sets and measures can be adapted to compute the Hausdorff dimension of the Rauzy gasket.

16:45 - 17:45 - Gabriel Fuhrmann (Durham)

Title: Prime periods on the interval

Abstract: Given two continuous self-maps f and g on the interval which have all periodic orbits in common (that is, O(x)={x,f(x),...,f^(p-1)(x)} is a p-periodic orbit of f if and only if it is a p-periodic orbit of g but a priori, f may permute the elements of O(x) in a different fashion than g does), it is natural to ask whether f=g on the closure of the periodic points (which is known to coincide with the closure of the recurrent points!). We show this is the case wherever orbits with prime periods are dense. Specifically, we show that mixing interval maps are uniquely determined by (the location of) their periodic orbits.

Some support is available for participants and any queries regarding support should be sent to Thomas Jordan, Thomas.Jordan@bristol.ac.uk.

For full details on previous editions of the Ergodic Theory Meeting, please visit the Past Meetings webpage.