How to show the geodesic orbit of a badly approximable number are/are not homogeneously equidistributed on its orbit closure?

Question

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 closed orbit equidistribution. By ergodicity, we already know this equidistribution holds for generic base points on the whole space but this fact is not helpful for this particular case as geodesic orbits of badly approximable numbers are not dense anyway.

Answer 1

A number is in BA if its orbit is bounded. Any such orbit closure must contain a full $A=\langle g_t\rangle$ orbit. By examining the possible subgroups, any such hypothetical $H$, as a stability group, must contain $A$ and can only be either $A$ or $B$ (the Borel subgroup) or all of $G$. As $B$ does not support a lattice, so cannot have a closed orbit, so if not $A$ it must be $G$. If the stability is whole of $G$, as $G$ acts transitively on $G/\Gamma$, the orbit is unbounded.

So we relaxed the problem of finding closed $A$ orbits (namely closed geodesics). This amounts to closed orbits of (real split) tori, which is well known to come from real quadratic fields.

In general, your hope of finding such non-trivial equidistribution results (pointwise) is slim as the action is Bernoulli (c.f. Katok's book), so in particular one can arrange an orbit closure of any arbitrary dimension...

P.S. You mentioned the pointwise ergodic theorem. It refers (in the standard formulation) to Haar-generic points, where in that case $H=G$, so in particular the orbit $H.x$ is unbounded (hence it does not tell us anything about BA points). Obviously you can apply the pointwise theorem for the case of a closed $A$ orbit (but then you are just having a translation over a torus and it is not really interesting).