Let $x\in \mathbb R$ be a badly approximable number. By Dani's correspondence, $g_t u_x \mathbb Z^2$ is bounded away from the cusp in $X=\text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$, identified with the space of unimodular lattices in $\mathbb R^2$. Here $g_t=\text{diag}(e^t,e^{-t})$ and $u_x=\begin{bmatrix} 1 & x\\ 0 & 1 \end{bmatrix}$.

I wonder if it is true that there exists a closed orbit $Hx_0$ where $H$ is a subgroup of $\text{SL}(2,\mathbb R)$ and a $H$-invariant Borel probability measure such that

$$\lim_{t\to \infty} \frac{1}{T}\int_0^T f(g_t u_x \mathbb Z^2)dt =\int_{Hx_0} f(y)d\mu_{Hx_0}(y)$$

for any $f\in C_c(X)$, and how to prove or disprove it. I am aware of Ratner's theory on unipotent flows but this is different. Sorry for my ignorance but the literature on this side is more on diophantine approximation rather than equidistribution. By ergodicity, we already know this equidistribution holds for generic base points.