= Solution
In a Hopf critical eigenspace, equip the real space $V\oplus V$ with the complex structure defined by the temporal phase action. A subgroup $\Sigma\subseteq G\times S^1$ is a <complex axial isotropy subgroup> when it is the full stabilizer of a nonzero vector and
$$
\boxed{\dim_{\mathbb C}\operatorname{Fix}(\Sigma)=1\quad\Longleftrightarrow\quad
\dim_{\mathbb R}\operatorname{Fix}(\Sigma)=2.}
$$
The isotropy is for the combined spatial and temporal action, so its elements encode <spatiotemporal symmetry of a periodic orbit>, not necessarily pointwise spatial symmetry. “Complex dimension” here refers to the Hopf complex structure; it need not agree with independently chosen spatial complex coordinates. Any kernel of the combined action belongs to every actual isotropy subgroup.
Back to article page
