Ergodic decompositions of geometric measures on Anosov homogeneous spaces (Q6561662)

Let \(G\) be a connected semisimple real algebraic group, \(P=MAN\) the Langlands decomposition of a minimal parabolic subgroup of \(G\) and \(P^0\) its identity component. Let \(\Gamma\) be a Zariski dense discrete subgroup \(\Gamma\in G\) which is Anosov with respect to \(P\). Fix a positive Weyl chamber \({\mathfrak a}^+\) in the Lie algebra \({\mathfrak a}\) of \(A\). Denote by \({\mathcal L}_\Gamma\subset{\mathfrak a}^+\) the limit cone of \(\Gamma\), write \(\psi_\Gamma:{\mathfrak a}\rightarrow{\mathbb R}\subset\{-\infty\}\) for the growth indicator function of \(\Gamma\) and define \N\[\ND_\Gamma^*:=\{\psi\in{\mathfrak a}^*\;\mid\;\psi\geq\psi_\Gamma,\;\psi(v)=\psi_\Gamma(v)\;\text{ for some }v\in\text{int }{\mathcal L}_\Gamma\}.\N\]\NFor \(\psi\in D_\Gamma^*\), denote by \(m_\psi^{\mathrm{BR}}\) (\(m_\psi^{\mathrm{BRS}}\)) the Burger-Roblin (Bowen-Margulis-Sullivan) measure on \(\Gamma\backslash G\). Let \({\mathfrak Y}_\Gamma\) be the collection of all \(P^0\)-minimal subsets of \(\Gamma\backslash G\) so that, fixing \({\mathcal E}_0\in {\mathfrak Y}_\Gamma\), one has a bijection \({\mathfrak Y}_\Gamma\simeq P/P_\Gamma\) where \(P_\Gamma=\{p\in P\;\mid\;{\mathcal E}_0p={\mathcal E}_0\}\). The main result proved in the paper under review can now be stated as follows: \vspace*{0.2cm}\N\NTheorem: For any Anosov subgroup \(\Gamma\subset G\) and \(\psi\in D_\Gamma^*\), one has:\N\begin{itemize}\N\item[(1)] the decomposition \(m_\psi^{\mathrm{BR}}=\sum_{{\mathcal E}_0\in {\mathfrak Y}_\Gamma}m_\psi^{\mathrm{BR}}\mid_{{\mathcal E}_0}\) is \(N\)-ergodic,\N\item[(2)] the decomposition \(m_\psi^{\mathrm{BRS}}=\sum_{{\mathcal E}_0\in {\mathfrak Y}_\Gamma}m_\psi^{\mathrm{BRS}}\mid_{{\mathcal E}_0}\) is \(A\)-ergodic,\N\item[(3)] the number of \(N\)-ergodic components of \(m_\psi^{\mathrm{BR}}\) and the number of \(A\)-ergodic components of \(m_\psi^{\mathrm{BRS}}\) are independent of \(\psi\) and equal the cardinality of \({\mathfrak Y}_\Gamma\).\N\end{itemize}