Given a plane billiard table with a smooth boundary which is a Jordan curve, I wonder if there is always a periodic orbit with sufficiently large period.
Formulation of my question: We are considering a dynamical billiard $B$ sitting in an Euclidean plane $\mathbb{R}^2$ and $\partial B$ is a smooth Jordan curve (possibly non-convex). For any positive integer $N$, is it true that there always exists an integer $n\ge N$ and an $n$-period trajectory? Here period $n$ means the closed orbit has $n$ bouncing points (the particle returns to the beginning position and direction only after $n$ times of reflection). Is there any $B$ such that all periodic orbits in $B$ have a uniform upper bound on their periods?
Some clues: For strictly convex $B$, a beautiful elementary argument due to Birkhoff asserts that there are $n$-periodic orbits for any $n\ge2$. See eg. this page or Theorem 6.2 of Tabachnikov's book. Also in arXiv:1409.5201 Arnaud proves that for a generic differentiable billiard table the periodic points are dense; it seems quite natural to believe the smooth members all have many periodic orbits.
A technical obstruction of generalizing the Birkhoff argument to non-convex smooth cases is that one cannot ensure that the critical point of the length function gives a legal trajectory that sits inside the interior of non-convex billiard table. But intuitively I guess this subtlety may be remedied by allowing the period (bouncing times) $n$ to be sufficiently large, hence the question.
