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We study the relation between critical exponents and Hausdorff dimensions oflimit sets for projective Anosov representations. We prove that the Hausdorffdimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times\mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponentsassociated respectively to a highest weight and a simple root.

Hausdorff dimension of limit sets for projective Anosov representations