Jordan and Cartan spectra in higher rank with applications to correlations

Abstract

For a given $d$-tuple $\rho=(\rho_1,\dots,\rho_d):\Gamma \to G$ 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 $\mathsf v=(v_1,\dots,v_d)$ in the {\it{spectrum cone}}, there exists $\delta_\rho(\mathsf v) > 0$ such that for any $\varepsilon_1, \dots, \varepsilon_d>0$, there exist $c_1,c_2> 0$ 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 {\it{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.