Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$.
Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in \overline{Kp}\setminus Kp$. A version of the Hilbert-Mumford criterion tells us that there is a rank-1 subtorus $\mathbb{C}^\times\cong H\subset K$ such that $q\in \overline{Hp}$. This is very nice, but I'm hoping for something stronger to hold.
Is it true that, given any decomposition $K = H_1\times\cdots H_\ell$ of $K$ into rank-1 subtori, there is some $i$ for which $q\in\overline{H_ip}$?
Unfortunately, I do not think this will hold. Let $T=H_1\times H_2$ with $H_i\cong\mathbb C^\times$ act on $\mathbb C^2$ component wise, i.e. $(t_1,t_2).(p_1,p_2):=(t_1p_1,t_2p_2)$. Consider the point $p=(1,1)$ and the point $q=(0,0)$. Let $H=\{ (t,t) \mid t\in\mathbb C^\times\}\subseteq T$ be the diagonal. Clearly, $q\in\overline{H.p}=\{ (t,t) \mid t\in\mathbb C\}$ but we also have $q\notin\overline{T_2.p}=\{1\}\times\mathbb C$ and $q\notin\overline{T_1.p}=\mathbb C\times\{1\}$.