Identities for hyperconvex Anosov representations

Abstract

In this paper, we establish Basmajian's identity for $(1,1,2)$-hyperconvex Anosov representations from a free group into $PGL(n, R)$. We then study our series identities on holomorphic families of Cantor non-conformal repellers associated to complex $(1,1,2)$-hyperconvex Anosov representations. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit nontrivial monodromy.