Non-Rigidity of Cyclic automorphic orbits in Free Groups

We say a subset Σ ⊆ FN of the free group of rank N is spectrally rigid if whenever T1, T2 ∈ cvN are R-trees in (unprojectivized) outer space for which ‖σ‖T1 = ‖σ‖T2 for every σ ∈ Σ, then T1 = T2 in cvN . The general theory of (non-abelian) actions of groups on R-trees establishes that T ∈ cvN is uniquely determined by its translation length function ‖ · ‖T : FN → R, and consequently that FN itself is spectrally rigid. Results of Smillie and Vogtmann [11], and of Cohen, Lustig, and Steiner [6] establish that no finite Σ is spectrally rigid. Capitalizing on their constructions, we prove that for any Φ ∈ Aut(FN ) and g ∈ FN , the set Σ = {Φ (g)}n∈Z is not spectrally rigid.