  For any strictly convex planar domain $\Omega \subset \mathbb{R}^2$ with a
$C^\infty$ boundary one can associate an infinite sequence of spectral
invariants introduced by Marvizi-Merlose. These invariants can generically be
determined using the spectrum of the Dirichlet problem of the Laplace operator.
A natural question asks if this collection is sufficient to determine $\Omega$
up to isometry. In this paper we give a counterexample, namely, we present two
non-isometric domains $\Omega$ and $\bar \Omega$ with the same collection of
Marvizi-Melrose invariants. Moreover, each domain has countably many periodic
orbits $\{S^n\}_{n \geqslant 1}$ (resp. $\{ \bar S^n\}_{n \geqslant 1}$) of
period going to infinity such that $ S^n $ and $ \bar S^n $ have the same
period and perimeter for each $ n $.
