Abstract. Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out(Γ)-invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group Γ into SLm(R). In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil-Petersson metric on the Fuchsian loci. Moreover, we produce Out(Γ)-invariant metrics on deformation spaces of convex cocompact representations into PSL 2 (C) and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group.

We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud-Guichard-Kassel-Wienhard [GGKW] and Kapovich-Leeb-Porti [KLP 2 ] by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz [ABY] and Bochi-Gourmelon [BG]. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti [KLP 2 ]. Date: June, 2017 (this version).