Non-Rigidity of Cyclic Automorphic Orbits in Free Groups

We say a subset $\Sigma \subseteq F_N$ of the free group of rank $N$ is \emph{spectrally rigid} if whenever $T_1, T_2 \in \cv_N$ are $\mathbb{R}$-trees in (unprojectivized) outer space for which $|\sigma|_{T_1} = |\sigma|_{T_2}$ for every $\sigma \in \Sigma$, then $T_1 = T_2$ in $\cv_N$. The general theory of (non-abelian) actions of groups on $\mathbb{R}$-trees establishes that $T \in \cv_N$ is uniquely determined by its translation length function $|\cdot|_T \colon F_N \to \mathbb{R}$, and consequently that $F_N$ itself is spectrally rigid. Results of Smillie and Vogtmann \cite{MR1182503}, and of Cohen, Lustig, and Steiner \cite{MR1105334} establish that no finite $\Sigma$ is spectrally rigid. Capitalizing on their constructions, we prove that for any $\Phi \in \Aut(F_N)$ and $g \in F_N$, the set $\Sigma = {\Phi^n(g)}_{n \in \mathbb{Z}}$ is not spectrally rigid.