In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent. In the appendix, in collaboration with M. Bridgeman, we extend a classical result on the Hessian of the Hausdorff dimension on purely imaginary directions. Contents 42 Appendix A. Continuity of entropy for non-Archimedean Anosov representations 46 Appendix B. Hessian of Hausdorff dimension on pure imaginary directions 49 References 54 (Cited on pages 5 and 35.

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into SL(n, R) that satisfy partial hyperconvexity properties inspired from Labourie's work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ-positive representations into SO(p, q) if p ≥ 4. We moreover show that 'positivity properties' known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations. Contents1. Introduction 2 2. Preliminaries 7 2.1. Anosov representations 8 3. Cross ratios 12 3.1. Projective cross ratios 12 3.2. Grassmannian cross ratio 13 3.3. Relations of the cross ratios 15 4. Partial hyperconvexity 15 4.1. Property H k 15 4.2. Property C k 18 4.3. Reducible representations and Fuchsian locii 22 4.4. Θ-positive representations 23 5. Positively ratioed representations 26 6. Proof of the collar lemma 28 7. A counterexample to the strong collar lemma 36 References 39

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into S L ( n , R ) SL(n,\mathbb {R}) that satisfy partial hyperconvexity properties inspired from Labourie’s work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ \Theta -positive representations into S O ( p , q ) SO(p,q) if p ≥ 4 p\geq 4 . We moreover show that ‘positivity properties’ known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.

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We extend classical results of Bridgeman-Taylor and McMullen on the Hessian of the Hausdorff dimension on quasi-Fuchsian space to the class of p1, 1, 2q-hyperconvex representations, a class introduced in [38] which includes small complex deformations of Hitchin representations and of Θ-positive representations. We also prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ Ñ POpn, 1q Ñ PUpn, 1q is positive definite when Γ is co-compact in POpn, 1q (unless n " 2 and the deformation is tangent to X `Γ, POp2, 1q ˘).