In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that mondromy of Margulis Space Times are "Anosov representations in non semi-simple Lie groups".

This paper defines the pressure metric on the Moduli space of Margulis spacetimes without cusps and shows that it is positive definite on the constant entropy sections. It also demonstrates an identity regarding the variation of the cross-ratios.Date: September 12, 2018.

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\textsf {SO}^0(n+1,n)\ltimes {\mathbb {R}}^{2n+1}$. We then show that a representation $\rho $ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt {L}_\rho $ is Anosov in $\textsf {SO}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $\rho (\Gamma )$ acts properly on $\mathbb {R}^{2n+1}$.

In this article we construct the pressure forms on the moduli spaces of higher dimensional Margulis spacetimes without cusps and study their properties. We show that the Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectra. We use this fact to show that the restrictions of the pressure form give Riemannian metrics on the constant entropy sections of the quotient moduli space. We also show that constant entropy sections of the moduli space with fixed linear parts bound convex domains.