The case of equality in the dichotomy of Mohammadi-Oh (Q2286620)

Let \(n \geq 2\) be an integer and \(G=\operatorname{SO}^{0}(1,n+1)\) be the group of direct isometries of the real \((n+1)\)-dimensional hyperbolic space \(\mathbf{H}^{n+1}\). \(G\) acts conformally on the boundary \(\partial\mathbf{H}^{n+1}\). We recall that the Busemann function is given by \[ b_{\xi}(x,y)=\lim_{t \longrightarrow +\infty}(d(x,\xi_t)-d(y,\xi_t)),\, \, \xi \in \partial\mathbf{H}^{n+1}, x,y \in \mathbf{H}^{n+1}, \] where \(t \mapsto \xi_t\) is some geodesic with positive endpoint \(\xi\). Let \(\Gamma\) be a non-elementary, Zariski-dense and convex-cocompact, discrete subgroup of \(G\). As customary, the critical exponent of \(\Gamma\) is denoted by \(\delta_{\Gamma}\) and it is given by \[ \delta_{\Gamma}=\limsup_{R \longrightarrow +\infty}\frac{\log(\text{Card}\big\{\gamma \in \Gamma~~:~~d(x,\gamma x) \leq R\})}{R}. \] which does not depend on the fixed point \(x \in \mathbf{H}^{n+1}\). We denote by \(N\) be a maximal unipotent subgroup of \(G\) (\(N\) is isomorphic to \(\mathbb{R}^n).\) We recall that the Bowen-Margulis-Sullivan (BMS) measure on the unit tangent bundle and on the frame bundle. Let \(T^1\mathbf{H}^{n+1}\) be the unit tangent bundle over \(\mathbf{H}^{n+1}.\) The Hopf map is the bijective mapping from \(T^1\mathbf{H}^{n+1} \times (\partial\mathbf{H}^{n+1} \times \partial\mathbf{H}^{n+1}\setminus \Delta) \times \mathbb{R}\) that maps the unit tangent vector \(u\) with base point \(x\) to the triple \((\xi,\eta,s)=(u^{-},u^{+},b_{u^{-}}(x,o))\), where \((u^{-},u^{+}\)respectively are the negative and positive endpoints of the geodesic whose derivative at \(t=0\) is \(u\) and \(\Delta\) is the diagonal set. In these coordinates, the BMS measure on \(T^1\mathbf{H}^{n+1}\) is given by \[ d\widetilde{\text{m}}_{\text{BMS}}(u) =e^{\delta_{\Gamma}b_{\xi}(x,u)+b_{\eta}(x,u)}d\mu_x(\xi)d\mu_x(\eta)ds, \] where the family \((\mu_x)_{x \in \mathbf{H}^{n+1} }\) of finite Borel measures on \(\partial\mathbf{H}^{n+1}\) satisfy: \begin{itemize} \item[1.] \(\Gamma\)-equivariance: \(\mu_{\gamma x}\) is the push-forward of \(\mu_{\gamma x}\) through the mapping induced by \(\gamma\) on \(\partial\mathbf{H}^{n+1}\). \item[2.] \(\delta_{\Gamma}\)-conformality: for any \(x,y \in \mathbf{H}^{n+1}\), \(\mu_x\) and \(\mu_y\) are equivalent and the Radon-Nikodym cocycle is given by \[ \frac{d\mu_y}{d\mu_x}(\xi)=e^{-\delta_{\Gamma} b_{\xi}(y,x)}, \] almost everywhere. \end{itemize} In the same manner, we define the Burger-Roblin (BR) measure by \[ d\widetilde{\text{m}}_{\text{BR}}(u) =e^{\delta_{\Gamma}b_{\xi}(x,u)+n b_{\eta}(x,u)}d\mu_x(\xi)d\nu_x(\eta)ds, \] where \(\nu_x\) is the unique Borel probability measure on \(\partial\mathbf{H}^{n+1}\) that is invariant under the stabilizer of \(x\) in \(G\). The BMS measure is a Radon measure that is invariant under the geodesic flow as well as under the natural operation of \(\Gamma\). The quotient of this measure with respect to \(\Gamma\) is a Radon measure \(m_{\text{BMS}}\) on \(\partial\mathbf{H}^{n+1}\) that is still invariant with respect to the geodesic flow. Likewise, the Burger-Roblin measure is \(\Gamma\)-invariant and thus defines a Radon measure on \(\partial\mathbf{H}^{n+1}\). In this setting, the author proves the following theorem. Theorem. Assume that \(\Gamma\) is convex-cocompact and Zariski-dense. Let \(m\) be an integer, \(1 \leq m \leq n \). If \(\delta_{\Gamma}=n\), then the Burger-Roblin measure is recurrent with respect to any \(m\)-plane \(U\) in \(N\), that is, for BR-almost every \(x\), there is a compact \(K\) such that \[ \int_{U}{1}_K(xu) du=+\infty. \]