Lemma
Let \(X\) be a toric variety with torus \(T\). Suppose \([C,f,\vec x]\in \mathcal M\) is a stable map (logarithmic or otherwise) which is fixed by the induced action on \(\mathcal M\). Then \(f(p) \in X^T\) whenever \(p\) is a marked point, a node or a ramification point of \(C\).
Proof
Let \([C,f,\vec x]\) be a fixed point of the moduli space. For each \(t\in T\) (pretending we’re working with varieties) the point \([C, t\circ f, \vec x]\) is therefore isomorphic to \([C,f,\vec x]\). This means we have a diagram
for each \(t\), where \(\varphi_t\) is an automorphism.
Suppose first that \(p\in C\) is a marked point. By definition of a morphism of marked points, \(\varphi_t\) must take the \(i\)th mark to the \(i\)th mark, hence \(\varphi_t(p) = p\). Commutativity above then immediately implies that \(f(p) = tf(p)\) for each \(t\).
Now suppose \(p\) is a node, in which case \(\varphi(p)\) is also a node. There are finitely many nodes of \(C\), so the set \(A = \{f(\varphi_t(p))\}_{t\in T}\) iterating over all possible \(t\in T\) is a finite set. By commutativity above we must have that \[t^{-1}f(p) \in A\] which means the orbit of \(f(p)\) is finite. The only finite orbit in a toric variety is a fixed point, hence \(f(p)\) is fixed.
Any \(\varphi_t\) must send ramification points to ramification points, and since there are finitely many of these in \(C\), so we are done.