One-day ergodic theory meeting

 This meeting is part of the LMS Scheme 3 grant between Birmingham, Bristol, Durham, Exeter, Loughborough, Manchester, Open, Queen Mary, St. Andrews, and Warwick.

 Programme:

14:00-15:00 Jerome Carrand (Sorbonne)

15:00-16:00 Richard Sharp (Warwick)

16:00-16:30 Coffee break

16:30-17:30 Gary Froyland (New South Wales)

All talks will take place in DAV1101 in the Sir David Davies Building (building number 14, West Park, please click on the university MAP).

Lunch is between 12:30-13:45 and will take place on campus at Village Restaurant and Elvyn Richards Dinning (building number 26, Village Park, please click on the university MAP)

After the talks we plan to go for dinner at the Basin restaurant, which is 30-minute walk from the lecture room, see google.

Titles and abstracts 

Speaker: Jerome Carrand

Title: Equilibrium states and MME for Sinai billiard flow  

Abstract: A Sinai billiard is a type of dispersive billiard having some hyperbolicity. After a short survey, we present a result of existence and uniqueness of equilibrium measures for the collision map (under some assumptions), that is, measures maximizing a quantity involving the metric entropy and a given Hölder potential. Assuming a stronger (but generically satisfied) condition we obtain the upper semi-continuity of the metric entropy. In a joint work with Viviane Baladi and Mark Demers, we bootstrap from the existence and uniqueness result to obtain a measure of maximal entropy for the billiard flow. We will present how the equilibrium measure is constructed by pairing maximal eigenvectors of a weighted transfer operator acting on anisotropic Banach spaces.

Speaker: Richard Sharp

Title: Periodic orbits and homology of knot complements 

Abstract: Consider the geodesic flow over a compact negatively curved surface or, more generally, a homologically full Anosov flow on a compact 3-manifold. The distribution of periodic orbits in a prescribed homology class is well-understood and is Gaussian as the class varies. In contrast, if one takes a non-compact finite area hyperbolic surface (or orbifold) then the presence of cusps introduces anomalous (Cauchy) behaviour. In this talk, we will address a variant problem, where extra homology is created by removing a finite number of (null-homologous) periodic orbits of the flow. There are examples where a homology fixed class in this complement will contain only finitely many periodic orbits but it is always possible to obtain asymptotics for periodic orbits whose homology grows in an appropriate linear fashion. This is joint work with Solly Coles.

Speaker: Gary Froyland

Title: Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory

Abstract: We consider quasi-compact linear operator cocycles driven by an invertible ergodic process and small perturbations of this cocycle. We prove an abstract pathwise first-order formula for the leading Lyapunov multipliers;  this result does not rely on random driving and applies also to sequential dynamics. We then consider the situation where the linear operator cocycle is a weighted transfer operator cocycle induced by a random map cocycle. The perturbed transfer operators are defined by the introduction of small random holes, creating a random open dynamical system. We obtain a first-order perturbation formula for the Lyapunov multipliers in this setting. Our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics with general ergodic invertible driving, and random observations. Further, in the setting of random piecewise expanding interval maps, we establish the existence of random equilibrium states and conditionally invariant measures for random open systems via a random perturbative approach. Finally we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We will illustrate the theory with some explicit examples.