Dynamics of abelian subgroups of GL(n, C): a structure's Theorem

In this paper, we characterize the dynamic of every abelian subgroups $\mathcal{G}$ of GL($n$, $\mathbb{K}$), $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that there exists a $\mathcal{G}$-invariant, dense open set $U$ in $\mathbb{K}^{n}$ saturated by minimal orbits with $\mathbb{K}^{n}- U$ a union of at most $n$ $\mathcal{G}$-invariant vectorial subspaces of $\mathbb{K}^{n}$ of dimension $n-1$ or $n-2$ on $\mathbb{K}$. As a consequence, $\mathcal{G}$ has height at most $n$ and in particular it admits a minimal set in $\mathbb{K}^{n}-\{0\}$.Comment: 16 page

Unique ergodicity of the horocycle flow on Riemannnian foliations

International audienceA classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations

Spectroscopic study of CdTe layers grown by molecular-beam epitaxy on (001) and (111) Cd0.96Zn0.04Te substrates

International audienceThe optical properties of CdTe grown by molecular-beam epitaxy are investigated by means of high-resolution photoluminescence, reflectivity, transmission, and resonant excitation spectroscopy. The CdTe epilayers are grown on (001)- and (111) B-oriented Cd0.96Zn0.04Te substrates. Sharp and strong lines associated with impurity bound exciton recombination dominate the spectra and indicate a good crystalline quality as well as a low level of impurity contamination. For the donor bound exciton lines, two electron transitions are observed which allow an identification of the chemical nature of the donors (probably Ga) by comparison with the data previously obtained on bulk material. For the acceptor lines, however, the shallow states which contribute to the spectra are different from those reported in the bulk and seem correlated with more complex centers. This study also reveals the importance of the residual strain contribution. Moreover, the marked difference observed for the optical properties between the (001) and the (111) epilayers spectra clearly evidence the change in the incorporation rate of impurities and/or grown-in defects density with the growth direction. The best spectra are obtained for the (001)-oriented CdTe layers

Growth gap in hyperbolic groups and amenability

International audienceWe prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane