Area-preserving twist maps have at least two different (p,q)-periodic
orbits and every (p,q)-periodic orbit has its (p,q)-periodic action for
suitable couples (p,q). We establish an exponentially small upper bound for
the differences of (p,q)-periodic actions when the map is analytic on a
(m,n)-resonant rotational invariant curve (resonant RIC) and p/q is
"sufficiently close" to m/n. The exponent in this upper bound is closely
related to the analyticity strip width of a suitable angular variable. The
result is obtained in two steps. First, we prove a Neishtadt-like theorem, in
which the n-th power of the twist map is written as an integrable twist map
plus an exponentially small remainder on the distance to the RIC. Second, we
apply the MacKay-Meiss-Percival action principle.
We apply our exponentially small upper bound to several billiard problems.
The resonant RIC is a boundary of the phase space in almost all of them. For
instance, we show that the lengths (respectively, areas) of all the
(1,q)-periodic billiard (respectively, dual billiard) trajectories inside
(respectively, outside) analytic strictly convex domains are exponentially
close in the period q. This improves some classical results of Marvizi,
Melrose, Colin de Verdi\`ere, Tabachnikov, and others about the smooth case