On the widths of the Arnol'd tongues

Let F : R -> R be a real analytic increasing diffeomorphism with F - Id being 1-periodic. Consider the translated family of maps. (F-t : R -> R)(t is an element of R) defined as F-t(x) = F(x) + t. Let Trans(F-t) be the translation number of F-t defined by Trans(F-t) := lim(n ->+infinity) F-t(on) - Id/n. Assume that there is a Herman ring of modulus 2 tau associated to F and let p(n)/q(n) be the nth convergent of Trans(F) = alpha is an element of R\Q. Denoting by l(theta) the length of the interval {t is an element of R vertical bar Trans(F-t) = theta}, we prove that the sequence (l(pn/qn)) decreases exponentially fast with respect to q(n). More precisely, lim sup(n ->+infinity) 1/q(n) log l(pn/qn) <= -2 pi tau. There is a relation between l(pn/qn) and the width of the Arnol'd tongue, which confirms that the widths of the tongues decrease exponentially fast under suitable conditions.