Affine and one-dimensional dynamics

Scientific program

The conference will consist of 4 mini-courses. And some talks.

Denjoy/Sacksteder theory Bertrand Deroin

We will discuss the role played by an important property called "Compact Generation", in the theory of 1 dimensional dynamical systems. Under this property, we will state and prove a theorem by Sacksteder, from the sixties, about the existence of hyperbolic fixed point in exceptional minimal sets. This result fails if compact generation is not satisfied, e.g. for affine interval exchanges.

TBA Isabelle Liousse

Abstract to come

On Cherry flows Liviana Palmisano

Answering to a conjecture of Poincaré, in 1936, Cherry provided the first example of a smooth flow on the two-dimensional torus for which the set containing all relevant and non-trivial dynamical information is itself non-trivial. Such a set is called a quasi-minimal set. In the mini-course we will present topological, metrical and ergodic properties of the quasi-minimal set which will be deduced by the study of the first return map to the flow.

Groups of affine and piecewise affine homeomorphisms Michele Triestino Notes

The translation T:x-->x+1 and the multiplication D:x-->2x generate a group of affine homeomorphisms of the real line, usually named BS(1,2) after Baumslag-Solitar, or dyadic affine group. This group is generated by the elements T,D, with the only relation DTD^{-1}=T^2.
This relation is definitely good from a dynamical point of view, in the sense that for any action, the dynamics of T is conjugate to the action of the squate T^2. Starting from is, the continuous actions of BS(1,2) on the real line are now well understood (Guelman-Liousse). It turns out that they are even more rigid (Bonatti-Monteverde-Navas-Rivas).
Introducing a third element H, coinciding with D in restriction to the positive half-line, at with the identity in restriction to the negative half-line, on finds an interesting group acting (faithfully) C^0 but not C^1 on the line (Bonatti-Lodha-T.). This group embeds in the recent examples of Monod, Lodha and Tatch-Moore of nonamenable groups without free subgroups.

Groupes d'homéomorphismes affines et affines par morceaux

La translation T:x-->x+1 et la multiplication D:x-->2x engendrent un groupe d'homéomorphismes affines de la droite réelle, souvent nommé BS(1,2) d'après Baumslag-Solitar, ou groupe affine dyadique. Ce groupe est engendré par les deux éléments T,D, avec la seule relation DTD^{-1}=T^2.
Cette relation se prête assez bien à l'étude des actions de ce groupe. Les actions C^0 de BS(1,2) sur la droite sont bien comprises (Guelman-Liousse). Il s'avère que les actions C^1 sont encore plus rigides (Bonatti-Monteverde-Navas-Rivas).
Si on introduit un troisième élément H qui coïncide avec D sur la demi-droite positive, et avec l'identité sur la demi-droite négative, on obtient un groupe intéressant qui agit C^0 mais pas C^1 sur la droite (fidèlement) (Bonatti-Lodha-T.). En outre on retrouve ce groupe dans les exemples récents de Monod, Lodha et Tatch Moore de groupes non-moyennables sans sous-groupes libres.

Full families of generalized Interval Exchange Transformations Luca Marchese

We consider generalized interval exchange transformations, or briefly GIETs, that is bijections of the interval which are piecewise increasing homeomorphisms with finite branches. When all continuous branches are translations, such maps are classical interval exchange transformations, or briefly IETs. The well-known Rauzy renormalization procedure extends to a given GIET and a Rauzy renormalization path is defined, provided that the map is infinitely renormalizable. We define full families of GIETs, that is optimal finite dimensional parameter families of GIETs such that any prescribed Rauzy renormalization path is realized by some map in the family. In particular, a GIET and a IET with the same Rauzy renormalization path are semi-conjugated. This extends a classical result of Poincar\'e relating circle homeomorphisms and irrational rotations. This is a joint work with Liviana Palmisano.

Schedule

Date	Time	Event
4 Monday December, 2017	9:30 - 10:30	Groups of affine and piecewise affine homeomorphisms - Michele Triestino
	11:00 - 12:00	Isabelle Liousse
	14:00 - 15:00	On Cherry flows - Liviana Palmisano
	15:30 - 16:30	Juliette Bavard
5 Tuesday December, 2017	9:30 - 10:30	On Cherry flows - Liviana Palmisano
	11:00 - 12:00	Isabelle Liousse
	14:00 - 15:00	Denjoy/Sacksteder theory - Bertrand Deroin
	15:30 - 16:30	Full families of generalized Interval Exchange Transformations - Luca Marchese
6 Wednesday December, 2017	9:30 - 10:30	Denjoy/Sacksteder theory - Bertrand Deroin
	11:00 - 12:00	Denjoy/Sacksteder theory - Bertrand Deroin
	14:00 - 20:00	Free afternoon
	20:00	Fondue evening
7 Thursday December, 2017	9:30 - 10:30	Groups of affine and piecewise affine homeomorphisms - Michele Triestino
	11:00 - 12:00	On Cherry flows - Liviana Palmisano
	14:00 - 15:00	Isabelle Liousse
	15:30 - 16:30	Adrien Boulanger
8 Friday December, 2017	9:30 - 10:30	Groups of affine and piecewise affine homeomorphisms - Michele Triestino
8 Friday December, 2017	11:00 - 12:00	Andy Sanders