Jordan and Cartan spectra in higher rank with applications to correlations

Abstract For a given $d$ -tuple $\rho =(\rho _1,\ldots ,\rho _d):\Gamma \to G$ of faithful Zariski-dense convex-cocompact representations of a finitely generated group $\Gamma$ , we study the correlations of length spectra $\{\ell _{\rho _i(\gamma )}\}_{[\gamma ]\in [\Gamma ]}$ and correlations of displacement spectra $\{\mathsf{d}(\rho _i(\gamma )o,o)\}_{\gamma \in \Gamma }$ . We prove that for any interior vector $\mathsf v=(v_1,\ldots ,v_d)$ in the spectrum cone , there exists $\delta _\rho (\mathsf v) \gt 0$ such that for any $\varepsilon _1, \ldots , \varepsilon _d\gt 0$ , there exist $c_1,c_2\gt 0$ such that \begin{align*} & \#\{[\gamma ]\in [\Gamma ]: v_iT \le \ell _{\rho _i(\gamma )} \le v_i T+\varepsilon _i, \;1 \le i \le d \} \sim c_1 \frac {e^{\delta _\rho ({\mathsf v})T}}{ T^{{(d+1)}/{2}}};\\ & \#\{\gamma \in \Gamma : v_iT \le \mathsf{d}(\rho _i(\gamma )o,o) \le v_i T+\varepsilon _i, \;1 \le i \le d \} \sim c_2 \frac {e^{\delta _\rho (\mathsf v)T}}{ T^{{(d-1)}/{2}}}. \end{align*} We deduce this result as a special case of our main theorem on the distribution of Jordan projections with holonomies and Cartan projections in tubes of an Anosov subgroup $\Gamma$ of a semisimple real algebraic group $G$ . We also show that the growth indicator of $\Gamma$ remains the same when we use Jordan projections instead of Cartan projections and tubes instead of cones, except possibly on the boundary of the limit cone.