Dynamical Spectral rigidity among $\mathbb Z_2$-symmetric strictly   convex domains close to a circle
  We show that any sufficiently (finitely) smooth $\mathbb Z_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak. 
Jacopo De Simoi, Vadim Kaloshin, Qiaoling Wei, with an appendix joint   with Hamid Hezari
math.DS math.SP