Renormalisation and limit theorems in ergodic theory (VALET=Vershik's automorphisms, limits in ergodic theory) – VALET

The project ``Renormalisation and limit theorem in ergodic theory'' is a project in Mathematics.
Having a core in dynamics, this project is located at the frontier between dynamics, geometry, algebraic geometry, combinatorics, and representation theory.

Consider an ergodic measure-preserving flow on a probability space. We can associate to this flow a random variable by considering the ergodic integral. A limit theorem of ergodic theory concerns the behavior of weak accumulation points of the normalized random variable.
For hyperbolic and partially hyperbolic dynamical systems these problems have been extensively studied. Much less is known for parabolic dynamical systems. The key tool to study these problems is renormalization.
In this project, we plan to study different flows and their renormalization in order to obtain limit theorems.
We have split the tasks in three parts. In the first one, we consider dynamical systems where the renormalization theory is not well understood. In the second part, we concentrate on Teichmuller flow. Finally we study ergodic action of groups.

One of the unifying links between the various parts of the project is the
fundamental concept of a Vershik's automorphism (also called ``adic
transformation''). Indeed, by the Vershik-Livshits theorem, substitutions
are naturally and explicitly isomorphic to Vershik's transformations, and
a similar construction can be given for tiling systems; furthermore,
as shown by the project coordinator, translation flows on flat surfaces
admit a natural representation as suspension flows over Vershik's
automorphisms corresponding to Bratteli diagrams given by Rauzy-Veech
renormalisation ; and, finally, the proposed study
of ergodic measures on matrix spaces is naturally formulated in terms of
the so-called ``graph of spectra'' and its canonical tail equivalence
relation.

The objective depends on the task. The first one concerns systems obtained by a Z action or by a Z^2 action. For these systems the symbolic dynamics can be useful in order to obtain new results. We investigate three systems: the billiard map, the piecewise isometries and some tilings spaces associated to multidimensional subshifts.


In the second task, the goal is to continue the study of Teichmuller flow in three directions:
the study of moduli space; the computation of Lyapunov exponents, the pseudo Anosov maps. For translation flows, the renormalizing flow is the Teichmuller flow on the moduli space of abelian differentials.

In the third task we consider group action on different spaces: the mapping class group, the horocycle flow and the measure on spaces of infinite matrices.


The problems which we plan work on are very ambitious and require consolidation of different approaches. We hope that the well-coordinated skills dispatched between different partners will allow us to attack these problems on a large front.

Ps: VALET means Vershik automorphisms, limits in ergodic theory.