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.