Ergodic properties of some linear actions (Q1848084)

Let \(\Gamma\) be a discrete group of \((2\times 2)\)-matrices with determinant 1 such that the quotient \(\Gamma\setminus \text{SL}(2,\mathbb{R})\) is compact and let \(\rho:\Gamma\to \mathbb{Z}^d\) be a representation of \(\Gamma\) on \(\mathbb{Z}^d\). The author surveys some ergodic properties of the linear action of \(\overline{\Gamma}= \ker\rho\) on \(\mathbb{R}^2\). Let \(L\) be the Lebesgue measure on \(\mathbb{R}^2\). The author proves, among other things, the following: Assume \(-Id\in\Gamma\). Then there is a function \(A: (\mathbb{R}^2\setminus \{0\})\times \mathbb{R}^+\to \mathbb{R}^+\) such that for any continuous function \(f\) with compact support on \(\mathbb{R}^2\setminus \{0\}\), we have for \(L\)-almost every \(x\in \mathbb{R}^2\), \[ \lim_{T\to\infty} \Biggl( \sum_{\gamma\in \overline{\Gamma}, \|\gamma\|< T} f(Jx)/ A(x,T)\Biggr)= \int(f(y)/|y|) dL(y) \] and \[ \limsup_{T\to\infty} A(x,T) (\log T)^{d/2}/T= C|x|^{-1} \] for some constant \(C\). Moreover, the constant \(C\) is given by a precise formula. The proof goes through a precise estimate of the sum \(\sum_{\gamma\in \overline{\Gamma},\|\gamma\|< T} f(Jx)\) for sufficiently many functions \(f\).