Billiards with a given number of (k,n)-orbits

We consider billiard dynamics inside a smooth strictly convex curve. For each pair of integers (k,n), we focus our attention on the billiard trajectory that traces a closed polygon with n sides and makes k turns inside the billiard table, called a (k,n)-orbit. Birkhoff proved that a strictly convex billiard always has at least two (k,n)-orbits for any relatively prime integers k and n such that 1≤k<n. In this paper, we show that Birkhoff’s lower bound is optimal by presenting examples of strictly convex billiards with exactly two (k,n)-orbits. We generalize the result to billiards with given even numbers of orbits for a finite number of periods.