*Speaker*: **Giulio Tiozzo** (University of Toronto, Canada)

*Title: ***Central limit theorems for counting measures in coarse negative curvature**

*Abstract:* We establish general central limit theorems for an action of a group on a hyperbolic space with respect to counting for the word length in the group.

In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018.

In our new work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces.

We will see how our techniques replace the classical thermodynamic formalism and allow us to provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.

Joint work with I. Gekhtman and S. Taylor.

**NOTE:** *The seminar will be streamed live on our YouTube channel then saved there. If you ask questions, with your video feed on or off, you agree to the use of your image/spoken words for said purpose.*

Prof. Heinz Hanssmann (University of Utrecht)

“Families of hyperbolic Hamiltonian tori”

Thursday 11 February 2016, 15.00

Dipartimento di Matematica (aula Dal Passo)

Università di Roma “Tor Vergata”

Abstract:

In integrable Hamiltonian systems hyperbolic tori form families,
parametrised by the actions conjugate to the toral angles. The
union over such a family is a normally hyperbolic invariant manifold.
Under Diophantine conditions a hyperbolic torus persists a small
perturbation away from integrability. Locally around such a torus
the normally hyperbolic invariant manifold is the centre manifold
of that torus and persists as well.

We are interested in `global’ persistence of the normally hyperbolic
invariant manifold. An important aspect is how the dynamics behaves
at the (topological) boundary. Where the manifold extends to infinity
this boundary is empty – this case makes clear that we need the
persistence theorem of normally hyperbolic invariant manifolds in
the non-compact setting.

If the normal hyperbolicity wanes as the boundary is approached we
need to ensure that the perturbed dynamics does not come closer
to the boundary. This provides the necessary uniform lower bound
of normal hyperbolicity to still ascertain persistence under small
perturbations. Making use of energy preservation and of Diophantine
tori persisting by KAM theory this can be achieved for families of
two-dimensional hyperbolic tori.

Thursday 12th November 2015, h 9:30

Sala Conferenze, Collegio Puteano, Centro De Giorgi, Pisa

Henk Bruin (University of Vienna)

“Sharp mixing rates via inducing with respect to general return times”

Abstract: For non-uniformly expanding maps inducing w.r.t. a general (i.e., not necessarily first) return time to Gibbs Markov maps, we provide sufficient conditions for obtaining sharp estimates for the correlation function. This applies to both the finite and the infinite measure setting. The results are illustrated by non-Markov intervals maps with an indifferent fixed point. This is joint work with Dalia Terhesiu.

Thursday 03/9/2015, h 16:30

Sala Conferenze (Puteano, Centro De Giorgi)

Mark Pollicott (University of Warwick)

“LINEAR RESPONSE AND PERIODIC ORBITS”

When: Wednesday July 22, 2015, at 11:30

Where: Seminario I, Dept. of Mathematics, Università di Bologna

Who: Francesco Cellarosi (Queen’s University, Canada)

What: Seminar “Recent progress towards Sarnak’s and Chowla’s Conjectures”

Abstract: I will present an overview of Sarnak’s conjecture on the disjointness of the Möbius function from any deterministic sequence and the related Chowla’s conjecture on the self-correlations of the Möbius function. Some progress towards weaker versions of these conjectures have been made recently, and I plan to illustrate them.