Ivrii's conjecture for some cases in outer and symplectic billiards

We give a proof for $(2n+1,n)$ and $(2n,n-1)$-periodic Ivrii’s conjecture for planar outer billiards. We also give new simple geometric proofs for the 3 and 4-periodic cases for outer and symplectic billiards, and generalize for higher dimensions in case of symplectic billiards.