# 2018-19

#### The seminars are held on Tuesdays at 14:00 in Room B3.02 - Mathematics Institute

(unless stated otherwise)

##### Organizers: Polina Vytnova and Selim Ghazouani

### Term 1

- 2 October

**Richard Sharp**(Warwick)

Title:*Periodic orbit growth on covers of Anosov flows*

Abstract: Suppose we have the lift of an Anosov flow to a regular cover. By considering periodic orbits intersecting some bounded region in the cover, one may define a Gurevic entropy which is less that or equal to the topological entropy of the Anosov flow. In the special case of a geodesic flow over a compact manifold with negative sectional curvatures, we have equality if and only if the cover is amenable. This result fails for other Anosov flows but we will discuss a natural generalisation. This is joint work with Rhiannon Dougall. - 9 October

**Michele Triestino**(Dijon)

Title:*Smoothening singular group actions on manifolds*

Abstract: Motivated by the recent results around Zimmer’s program, we study group actions on manifolds, with singular regularity (we require that every element is differentiable at all but countably many points). The groups under considerations have a fixed point property, named*FW*, which generalizes Kazhdan’s property (*T*) (in particular we can consider actions of lattices in higher rank simple Lie groups).

The main result is that if a group*G*has property*FW*, any singular action of*G*on a closed manifold

1) either has a finite orbit,

2) or is conjugate to a differentiable action, up to changing the differentiable structure of the manifold.

This is a joint work with Yash Lodha and Nicolas Matte Bon. - 16 October

**Etienne Le Masson**(Universite de Cergy-Pontoise)

Title:*Quantum ergodicity and random waves in the Benjamini-Schramm limit*

Abstract: One of the fundamental problems in quantum chaos is to understand how high-frequency waves behave in chaotic environments. A famous but vague conjecture of Michael Berry predicts that they should look on small scales like Gaussian random fields. We will show how the notion of Benjamini-Schramm convergence of manifolds (originally defined for graphs) can be used to formulate Berry’s conjecture precisely. The Benjamini-Schramm convergence includes the high-frequency limit as a special case but provides a much more general framework that will lead us to consider a case where the frequencies stay bounded and the size of the manifold increases instead. In this alternative setting we will explain how the ergodicity of the Gaussian wave and the mixing of the geodesic flow can be used to prove weaker forms or consequences of the random wave conjecture.

Joint work with Miklos Abert and Nicolas Bergeron. - 17 October (2:00pm, B3.02)

**Hillel Furstenberg**(Jerusalem)

Title:*Affine group actions*

Abstract: When*X*is a compact, convex space, and a group*G*acts on*X*preserving the affine structure, we speak of an "affine action", or, representation. In analogy with linear representation theory, one would like to describe all minimal — or irreducible — affine actions. We develop the theory for Lie groups and focus on*PSL(2,R)*, noting that every bounded harmonic function in the unit disc leads to an irreducible affine representation. It turns out surprisingly that up to equivalence, this group has a unique irreducible affine representation. - 23 October
**Stephen Cantrell**(Warwick)

Title:*Comparing word length with displacement for actions on CAT(-1) spaces*

Abstract: Suppose a hyperbolic group*G*acts sufficiently nicely on a complete CAT(-1) geodesic metric space*X*. Fix an origin for*X*. Each group element*g*in*G*displaces this origin by a distance comparable to the word length of*g*. In this talk we discuss various ways in which we can make an averaged comparison between the word length and displacement associated to the action of*G*on*X*. We will also discuss other geometrically interesting real valued functions on hyperbolic groups, for which these comparison results apply. - 30 October
**Dmitry Turaev**(Imperial)

Title:*On wandering domains near a homoclinic tangency*

**Gabriella Keszthelyi**(Rényi Institute, Budapest)

Title:*Dynamical properties of biparametric skew tent maps*

Abstract: pdf - 6 November
**Federico Rodriguez-Hertz**(Penn State/Lille) - 13 November
**Jose Alves**(Porto/Loughborough) - 20 November
**Ariel Rapaport**(Cambridge) - 27 November

**Björn Winckler**(Imperial)

Title:*Instability of renormalization*

Abstract: In this talk I will discuss renormalization of low-dimensional dynamical systems and associated phenomena such as universality and rigidity. The first part of the talk will be an introduction to the classical results in this field. In the second part I will discuss new phenomena that appear in the article 'Instability of Renormalization' (arXiv:1609.04473). In particular, I will introduce the renormalization operator acting on Lorenz maps (these are one-dimensional maps associated with three-dimensional flows undergoing a homoclinic bifurcation). This operator turns out to have non-trivial dynamics inside topological classes of stationary type which in turns leads to the phenomena of coexistence and as dimensional discrepancy. All of these notions will be explained in the talk. - 4 December
**Mike Whittaker**(Glasgow)

### Term 2

- 8 January
- 15 January
**Igors Gorbovitskis**(Bremen) - 22 January
- 29 January
**Victor Kleptsyn**(Rennes) - 5 February
- 12 February
**Yuri Lima**(UFC) - 19 February
- 26 February
- 5 March
**Jeroen Lamb**(Imperial) - 12 March