Jordan and Cartan spectra in higher rank with applications to correlations

Abstract

For a given $d$-tuple $ρ=(ρ_1,\dots,ρ_d):Γ\to G$ of faithful Zariski dense convex cocompact representations of a finitely generated group $Γ$, we study the correlations of length spectra $\{\ell_{ρ_i(γ)}\}_{[γ]\in[Γ]}$ and correlations of displacement spectra $\{\mathsf{d}(ρ_i(γ)o,o)\}_{γ\inΓ}$. We prove that for any interior vector $\mathsf v=(v_1,\dots,v_d)$ in the {\it{spectrum cone}}, there exists $δ_ρ(\mathsf v) > 0$ such that for any $\varepsilon_1, \dots, \varepsilon_d>0$, there exist $c_1,c_2> 0$ such that \begin{align*} &\#\{[γ]\in [Γ]: v_iT \le \ell_{ρ_i(γ)} \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_1 \frac{e^{δ_ρ(\mathsf{v})T}}{ T^{{(d+1)}/{2}}};\\ &\#\{γ\in Γ: v_iT \le \mathsf{d}(ρ_i(γ)o,o) \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_2 \frac{e^{δ_ρ(\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 $Γ$ of a semisimple real algebraic group $G$. We also show that the growth indicator of $Γ$ 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.