Deformative Magnetic Marked Length Spectrum Rigidity

Let $M$ be a closed surface and let $\{g_s\mathrm{\mid }s\in (-\epsilon ,\epsilon )\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{{\kappa }_s:M\to \mathbb{R}\mathrm{\mid }s\in (-\epsilon ,\epsilon )\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s,{\kappa }_s)\mathrm{\mid }s\in (-\epsilon ,\epsilon )\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s\in (-\epsilon ,\epsilon )$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s:M\to M\mathrm{\mid }s\in (-\epsilon ,\epsilon )\}$ such that $f_s^{\ast }(g_s)=g_0$ and $f_s^{\ast }({\kappa }_s)={\kappa }_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows.