Identities for hyperconvex Anosov representations

TL;DR

This paper extends Basmajian's identity to certain hyperconvex Anosov representations, analyzing series identities on Cantor sets and their analytic properties related to Hausdorff dimension.

Contribution

It establishes new series identities for hyperconvex Anosov representations and studies their analytic continuation and monodromy properties.

Findings

Series identities are absolutely summable when the Cantor set's Hausdorff dimension is less than one.

Identities can be analytically continued within their domain of convergence.

The identities exhibit nontrivial monodromy during continuation.

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.