# Conference - Young researcher meeting in dynamics and geometry

Rennes, from September 6th to September 8th

Organization board: Françoise Dal'Bo, Frédéric Paulin, Barbara Schapira, Damien Thomine

Since its creation the Platon network (GDR National Center for Scientific Research n°3341 http: // costia.free.fr / platon/) leads actions towards young researchers in ergodic geometry. The recurrent young researcher meeting is one of the highlights of the year. The goal is to allow about ten PhD students or recent doctors to expose their work and promotes discussions between young and senior researchers.
The "Young researcher meeting in dynamics and geometry" follows the spirit of these recurring meetings with an international dimension brought in particular by Swiss and Senegalese networks.

See the program and practical information here

TALKS

Alexander Adam (UPMC) Resonances for Anosov diffeomorphism

Kamel Belarif (Université de Bretagne Occidentale) Genericity of weak mixing in negative curvature

Filippo Cerocchi (Max Planck Institute for Mathematics, Bonn) Rigidity and finiteness for compact 3-manifolds with bounded entropy

Maria Cumplido Cabello (Université de Rennes 1) Loxodromic actions of Artin-Tits groups

Nguyen-Bac Dang (Ecole Polytechnique) Degrees of iterates of rational maps

Laurent Dufloux (Oulu University) Hausdorff dimension of limit sets at the boundary of the complex hyperbolic plane

Mikolaj Fraczyk (Université Paris-Sud) Mod p homology growth of locally symmetric spaces

Weikun He (Université Paris-Sud) Sum-product estimates and equidistribution of toral automorphisms

Cyril Lacoste (Université de Rennes 1) Dimension rigidity of lattices in semisimple Lie groups

Erika Pieroni (Università di Roma, Sapienza) Minimal Entropy of 3-manifolds

Fanni M. Selley (Budapest University of Technology) Ergodicity breaking in mean-field coupled map systems

Nasab Yassine (Université de Bretagne Occidentale) Quantitative recurrence of one-dimensional dynamical systems preserving an infinite measure

ABSTRACTS

• Alexander Adam Resonances for Anosov diffeomorphism

The deterministic chaotic behavior of an invertible map T is
appropriately described by the existence of expanding and contracting
directions of the differential of T. A special class of such maps
consist in Anosov diffeomorphisms. Every 2-by-2 hyperbolic matrix M
with integer entries induces such a diffeomorphism on the 2-torus.
For all pairs of real-analytic functions on the 2-torus, one defines a
correlation function for T which captures the asymptotic independence
of such a pair under the evolution T^n as n tends to infinity. What is
the rate of convergence of the correlation as n tends to infinity, for
instance what is its decay rate? The resonances for T are the poles of
the Z-transform of the meromorphic continued correlation function. The
decay rate is well-understood if T=M. There are no non-trivial
resonances of M. In this talk, I consider small real-analytic
perturbations T of M where at least one non-trivial resonance of T
appears. This affects the decay rate of the correlation.

• Kamel Belarif Genericity of weak mixing in negative curvature

Let M be a manifold with pinched negative sectional curvature. We show
that, when M is geometrically finite and the geodesic flow on T^1M is
topologically mixing, the set of mixing invariant measures is
dense in the set P(T^1M) of invariant probability measures. This
implies that the set of weak-mixing measures which are invariant by
the geodesic flow is a dense G-delta subset of P(T^1M). We also show how
to extend these results to geometrically infinite manifolds with cusps
or with constant negative curvature.

Avec un échange d'intervalle affine donné vient naturellement une
famille de telles dynamiques indexées par le cercle. En effet, la
pré-composition par une rotation de l'application initiale définit un
autre échange d'intervalle affine. On étudiera cette famille de
dynamiques dans un cas particulier à travers la géométrie de la
surface affine associée et son groupe de transformation affine.

An affine interval exchange (AIE) is a piecewise affine map from the
circle to itself. Such a map defines a dynamical systems over the
circle by iterating it. With an AIE comes naturally a family of AIE
indexed by the circle: they are defined by pre-composing the initial
AIE by a rotation. The presentation will focus on the study of
possible dynamical behaviors of such a family of AIE through a
peculiar example.

• Filippo Cerocchi Rigidity and finiteness for compact 3-manifolds with bounded entropy

We present some local topological rigidity results for the set S of
non-geometric, compact -- with possibly empty boundary and no
spherical boundary components --, orientable Riemannian 3-manifolds
having torsionfree fundamental group, with bounded entropy and
diameter. By "local", we mean that we consider S endowed with the
Gromov-Hausdorff-topology. We shall provide examples to show the
necessity of the assumptions and discuss some open problems.
Moreover, we shall give a proof of the finiteness of the homeomorphism
types of the manifolds in S. These are joint works with A. Sambusetti
(Rome, Sapienza).

• Maria Cumplido Cabello Loxodromic actions of Artin-Tits groups

Artin-Tits groups act on a certain delta-hyperbolic complex, called
the additional length complex". For an element of the group, acting
loxodromically on this complex is a property analogous to the property
of being pseudo-Anosov for elements of mapping class groups. A
well-known conjecture about mapping class groups claims that "most
elements" of the mapping class group of a surface are
pseudo-Anosov. In fact, we can prove that a positive proportion is
pseudo-Anosov.

By analogy, we conjecture that most'' elements of Artin-Tits groups
act loxodromically. More precisely, in the Cayley graph of a subgroup
G of an Artin-Tits group, the proportion of loxodromically acting
elements in a ball of large radius should tend to one as the radius
tends to infinity. We will give a condition guaranteeing that this
proportion stays away from zero. This condition is satisfied e.g. for
Artin-Tits groups of spherical type, their pure subgroups and some of
their commutator subgroups.

• N'Guyen-Bac Dang Degrees of iterates of rational maps

In this talk, I will explain what is a rational map, how to define its
k-degrees, and I will study the k-degrees of its iterates. I will
explain how the study of the growth of these sequences of numbers helps
in understanding the dynamics of these maps.

• Laurent Dufloux Hausdorff dimension of limit sets at the boundary of complex hyperbolic planes

Consider the standard contact structure on the 3-sphere. The
associated subriemannian metric has dimension 4. The Gromov comparison
subriemannian metric is related tothe Hausdorff dimension with respect to
the usual (Riemannian) metric. We will look at this problem in the
case of limit sets of discrete groups of complex hyperbolic
isometries.

• Mikolaj Fraczyk Mod p homology growth of locally symmetric spaces

I will talk about the growth of the dimension of mod-p homology groups
of locally symmetric spaces. Let G be a higher rank Lie group and X its
symmetric space and let L be a lattice in G. Results on the rank
gradient by Abert, Gelander and Nikolov imply that if L is right
angled then for every sequence of subgroups (L_n) of L, the dimensions
of the homology groups H_1(X/L_n,Z/pZ) grow sublinearly in the volume
of X/L_n. In the special case p=2, I showed that the same
statement holds for any sequence of lattices L_n with volume
escaping to infinity (even if they are pairwise non-commensurable).

• Weikun He Sum-product estimates and equidistribution of toral automorphisms

Bourgain's sum-product theorem is a metric version of ErdÅ‘s-SzemerÃ©di
sum-product theorem. It asserts that a typical set of real numbers
grows fast under addition and multiplication. We will present a
generalisation of Bourgain's theorem to matrix algebras and discuss
how it is motivated by a ergodic problem, namely, quantitative
equidistributions of orbits on the d-dimensional torus under
sub-semigroups of SL(d,Z).

• Cyril Lacoste Dimension rigidity of lattices in semisimple Lie groups

We study actions of discrete groups on classifying spaces (or
classifying spaces for proper actions). For instance the hyperbolic
plane is a classifying space for proper actions of the group PSL(2,Z)
(but not of minimal dimension). Such spaces can be used to compute the
cohomology of the group, so we want them to have the lowest possible
dimension. This leads us to the definitons of the (proper) geometric
dimension and the (virtual) cohomological dimension. These two
dimensions are not always equal, we will see it is the case for a
lattice in the group of isometries G of a symmetric space of
non-compact type without Euclidean factors (such a group is a
semisimple Lie group but not necessarily connected). This result has
an important consequence called "dimension rigidity", that is, the two
dimensions are still equal for a group commensurable to a lattice of
G.

• Erika Pieroni Minimal Entropy of 3-manifolds

We present the solution of the minimal entropy problem for non-geometric,
closed, orientable 3-manifolds (that is, those manifolds which do not admit a com-
plete metric locally isometric to one of the eight 3-dimensional model geometries).
Together with the results of Besson-Courtois-Gallot for locally symmetric spaces
and the work of Soma, Gromov et.al. on the simplicial volume of 3-manifolds and
its relation with entropy, this gives a complete picture of the minimal entropy prob-
lem for all closed, orientable 3-manifolds. Our work strongly builds on Souto's PhD
work (unpublished), filling some gaps in the proof and completing the picture in
the case of non-prime manifolds. In detail, we show that the minimal entropy is ad-
ditive with respect to the prime decomposition and that for an irreducible manifold
X it coincides with the sum of the volume entropies of all the
JSJ components of hyperbolic type, each endowed with its complete,
hyperbolic metric of nite volume. For the lower bound of MinEnt(X), we adapt
Besson-Courtois-Gallot's barycenter method following Souto's ideas; then, we show
how this lower bound is realized by producing a sequence of Riemannian metrics
gk on X whose volume-entropies tend to

• Fanni M. SelleyErgodicity breaking in mean-field coupled map systems

Coupled map systems are simple models of a finite or infinite network
of interacting units. The dynamics of the compound system is given by
the composition of the (typically chaotic) individual dynamics and a
coupling map representing the characteristics of the interaction. The
coupling map usually includes a parameter s in [0,1], representing the
strength of interaction. The main interest in such models lies in the
emergence of bifurcations when s is varied. We first introduce our
results for small finite systems. Then we initiate a new point of view
which focuses on the evolution of distributions and allows to
incorporate the investigation of a continuum of sites.

• Nasab Yassine Quantitative recurrence of one-dimensional dynamical systems preserving an infinite measure

We are interested in the asymptotic behaviour of the first return time
of the orbits of a dynamical system into a small neighbourhood of
their starting points. We study this quantity in the context of
dynamical systems preserving an infinite measure. More precisely, we
consider the case of Z-extensions of subshifts of finite type. We also
consider a toy probabilistic model in order to enlighten the strategy
of our proofs.