Invariant measures of the topological flow and measures at infinity on hyperbolic groups (Q6592861)

Let \(\Gamma\) be a non-elementary hyperbolic group, \(\partial \Gamma\) the ideal boundary of \(\Gamma\) equipped with a quasi-metric and let \(\partial^{2} \Gamma=(\partial \Gamma)^{2} \setminus \{ \mathrm{diagonal} \}\) be the boundary square endowed with the diagonal action by \(\Gamma\). If \(\kappa: \Gamma \times \partial^{2} \Gamma \rightarrow \mathbb{R}\) is a Hölder continuous cocycle coming from the Busemann function for a strongly hyperbolic metric, the authors define a compact space \(\mathcal{F}_{k}=\Gamma \setminus (\partial^{2} \Gamma \times \mathbb{R})\) equipped with an \(\mathbb{R}\)-action, that they call a topological flow (see Section 3 for a precise definition).\N\NThe purpose of the paper under review is to show that there exists a coding based on a transitive subshift of finite type for a flow associated with every non-elementary hyperbolic group. Applications include the real analyticity of Manhattan curves (introduced by \textit{M. Burger} in [Int. Math. Res. Not. 1993, No. 7, 217--225 (1993; Zbl 0829.57023)]), the uniqueness of measure of maximal Hausdorff dimension with potentials, and the real analyticity of intersection numbers for perturbations of dominated representations. This provides a direct proof of a result shown by \textit{M. Bridgeman} et al. in [Geom. Funct. Anal. 25, No. 4, 1089--1179 (2015; Zbl 1360.37078)]. The authors also apply their results to give a multifractal analysis of harmonic measures for finite range random walks and discuss, for explicit examples, the relationships between various measure classes on the boundary of the group.