CONTINUITY OF THE PHONON GAP

The phonon gap G for an invariant set LAMBDA of a symplectic twist map of R(d) X R(d) with action functional W is the infimum of \\D2W(x)(z)xi\\2 over z is-an-element-of LAMBDA and variations xi with \\xi\\2 = 1. It is proved here that if LAMBDA(k), k is-an-element-of N, is a sequence of compact invariant sets converging in Hausdorff topology to a compact invariant set LAMBDA, then G(LAMBDA(k)) converges to G(LAMBDA). The result implies that the phonon gap is an excellent quantifier of uniform hyperbolicity. Several generalisations are sketched.