M. Kac popularized the question `Can you hear the shape of a drum?'. Mathematically, consider a bounded planar domain Ω and the associated Dirichlet problem Δu + λ2u = 0 with u|∂Ω = 0. The set of λs such that this equation has a solution, denoted ℒ(Ω) is called the Laplace spectrum of Ω. Does Laplace spectrum determine Ω? In general, the answer is negative.
Consider the billiard problem inside Ω. Call the length spectrum the closure of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of the wave trace function, generically, the Laplace spectrum determines the length spectrum. We show that any axis symmetric planar domain with sufficiently smooth boundary close to the disk is dynamically spectrally rigid, i.e. can't be deformed without changing the length spectrum. This partially answers a question of P. Sarnak. This is a joint work with J. De Simoi and Q. Wei.